diff --git a/CHANGES.rst b/CHANGES.rst
index 40d6984ca55..d7305cf71ac 100644
--- a/CHANGES.rst
+++ b/CHANGES.rst
@@ -40,6 +40,9 @@ Bugs fixed
 * #13619: LaTeX: possible duplicated footnotes in PDF from object signatures
   (typically if :confval:`latex_show_urls` ``= 'footnote'``).
   Patch by Jean-François B.
+* #13621: linkcheck: Fix false "Anchor not found" errors for valid URLs with encoded
+  characters in fragment identifiers.
+  Patch by Eric Nordlund.
 
 Testing
 -------
diff --git a/sphinx/builders/linkcheck.py b/sphinx/builders/linkcheck.py
index ff6878f2acb..085acd5fe66 100644
--- a/sphinx/builders/linkcheck.py
+++ b/sphinx/builders/linkcheck.py
@@ -11,7 +11,7 @@
 from html.parser import HTMLParser
 from queue import PriorityQueue, Queue
 from threading import Thread
-from typing import TYPE_CHECKING, NamedTuple, cast
+from typing import TYPE_CHECKING, Any, NamedTuple, cast
 from urllib.parse import quote, unquote, urlparse, urlsplit, urlunparse
 
 from docutils import nodes
@@ -485,6 +485,9 @@ def _retrieval_methods(
 
     def _check_uri(self, uri: str, hyperlink: Hyperlink) -> _URIProperties:
         req_url, delimiter, anchor = uri.partition('#')
+        original_encoded_anchor = (
+            anchor  # Store the original encoded anchor before decoding
+        )
         if delimiter and anchor:
             for rex in self.anchors_ignore:
                 if rex.match(anchor):
@@ -536,7 +539,9 @@ def _check_uri(self, uri: str, hyperlink: Hyperlink) -> _URIProperties:
                 ) as response:
                     if anchor and self.check_anchors and response.ok:
                         try:
-                            found = contains_anchor(response, anchor)
+                            found = contains_anchor(
+                                response, anchor, original_encoded_anchor
+                            )
                         except UnicodeDecodeError:
                             return (
                                 _Status.IGNORED,
@@ -686,11 +691,13 @@ def _get_request_headers(
     return {}
 
 
-def contains_anchor(response: Response, anchor: str) -> bool:
+def contains_anchor(
+    response: Response, anchor: str, original_encoded_anchor: str = ''
+) -> bool:
     """Determine if an anchor is contained within an HTTP response."""
-    parser = AnchorCheckParser(anchor)
-    # Read file in chunks. If we find a matching anchor, we break
-    # the loop early in hopes not to have to download the whole thing.
+    parser = AnchorCheckParser(anchor, original_encoded_anchor)
+    # Read file in chunks. If we find a matching anchor, we break the loop early
+    # to avoid downloading the entire response body.
     for chunk in response.iter_content(chunk_size=4096, decode_unicode=True):
         if isinstance(chunk, bytes):
             # requests failed to decode, manually try to decode it
@@ -706,15 +713,37 @@ def contains_anchor(response: Response, anchor: str) -> bool:
 class AnchorCheckParser(HTMLParser):
     """Specialised HTML parser that looks for a specific anchor."""
 
-    def __init__(self, search_anchor: str) -> None:
+    def __init__(self, search_anchor: str, original_encoded_anchor: str = '') -> None:
+        """Initialize the parser with multiple anchor variations.
+
+        Args:
+            search_anchor: The decoded anchor to search for
+                (e.g., "standard-input/output-stdio")
+            original_encoded_anchor: The original encoded anchor
+                (e.g., "standard-input%2Foutput-stdio")
+        """
         super().__init__()
 
-        self.search_anchor = search_anchor
+        # Create variations of the anchor to check
+        self.search_variations = {
+            search_anchor,  # decoded (current behavior)
+        }
+
+        # Add the original encoded version if provided
+        if original_encoded_anchor:
+            self.search_variations.add(original_encoded_anchor)
+
+        # Add a re-encoded version if the decoded anchor contains characters
+        # that would be encoded
+        if search_anchor != quote(search_anchor, safe=''):
+            self.search_variations.add(quote(search_anchor, safe=''))
+
         self.found = False
 
     def handle_starttag(self, tag: Any, attrs: Any) -> None:
         for key, value in attrs:
-            if key in {'id', 'name'} and value == self.search_anchor:
+            # Check if the attribute value matches any of our variations
+            if key in {'id', 'name'} and value in self.search_variations:
                 self.found = True
                 break
 
diff --git a/tests/roots/test-linkcheck-encoded-anchors/conf.py b/tests/roots/test-linkcheck-encoded-anchors/conf.py
new file mode 100644
index 00000000000..88442504003
--- /dev/null
+++ b/tests/roots/test-linkcheck-encoded-anchors/conf.py
@@ -0,0 +1,4 @@
+root_doc = 'encoded_anchors'
+exclude_patterns = ['_build']
+linkcheck_anchors = True
+linkcheck_timeout = 0.25
diff --git a/tests/test_builders/test_anchor_check_parser.py b/tests/test_builders/test_anchor_check_parser.py
new file mode 100644
index 00000000000..c58a37158a7
--- /dev/null
+++ b/tests/test_builders/test_anchor_check_parser.py
@@ -0,0 +1,76 @@
+"""Test the AnchorCheckParser class."""
+
+from __future__ import annotations
+
+from typing import TYPE_CHECKING
+from unittest import mock
+
+from sphinx.builders.linkcheck import AnchorCheckParser, contains_anchor
+
+if TYPE_CHECKING:
+    from typing import Any
+
+
+def test_anchor_check_parser_basic() -> None:
+    """Test basic anchor matching functionality."""
+    parser = AnchorCheckParser('test-anchor')
+    parser.feed('<html><body><div id="test-anchor">Test</div></body></html>')
+    assert parser.found is True
+
+    parser = AnchorCheckParser('non-existent')
+    parser.feed('<html><body><div id="test-anchor">Test</div></body></html>')
+    assert parser.found is False
+
+
+def test_anchor_check_parser_with_encoded_anchors() -> None:
+    """Test anchor matching with encoded characters."""
+    # Test with encoded slash
+    parser = AnchorCheckParser(
+        'standard-input/output-stdio', 'standard-input%2Foutput-stdio'
+    )
+    parser.feed(
+        '<html><body><div id="standard-input%2Foutput-stdio">Test</div></body></html>'
+    )
+    assert parser.found is True
+
+    # Test with plus sign
+    parser = AnchorCheckParser('encoded+anchor', 'encoded%2Banchor')
+    parser.feed('<html><body><div id="encoded%2Banchor">Test</div></body></html>')
+    assert parser.found is True
+
+    # Test with space
+    parser = AnchorCheckParser('encoded space', 'encoded%20space')
+    parser.feed('<html><body><div id="encoded%20space">Test</div></body></html>')
+    assert parser.found is True
+
+
+def test_contains_anchor_with_encoded_characters() -> None:
+    """Test the contains_anchor function with encoded characters."""
+    mock_response = mock.MagicMock()
+
+    # Setup a response that returns HTML with encoded anchors
+    def mock_iter_content(chunk_size: Any = None, decode_unicode: Any = None) -> Any:
+        content = '<html><body><div id="standard-input%2Foutput-stdio">Test</div></body></html>'
+        yield content
+
+    mock_response.iter_content = mock_iter_content
+
+    # Test with original encoded anchor
+    assert (
+        contains_anchor(
+            mock_response,
+            'standard-input/output-stdio',
+            'standard-input%2Foutput-stdio',
+        )
+        is True
+    )
+
+    # Test with decoded anchor only
+    mock_response2 = mock.MagicMock()
+    mock_response2.iter_content = mock_iter_content
+    assert contains_anchor(mock_response2, 'standard-input/output-stdio') is True
+
+    # Test with non-existent anchor
+    mock_response3 = mock.MagicMock()
+    mock_response3.iter_content = mock_iter_content
+    assert contains_anchor(mock_response3, 'non-existent-anchor') is False
diff --git a/tests/test_builders/test_build_linkcheck_encoded_anchors.py b/tests/test_builders/test_build_linkcheck_encoded_anchors.py
new file mode 100644
index 00000000000..596d4e093b4
--- /dev/null
+++ b/tests/test_builders/test_build_linkcheck_encoded_anchors.py
@@ -0,0 +1,169 @@
+"""Test the linkcheck builder's ability to handle encoded anchors."""
+
+from __future__ import annotations
+
+import json
+import re
+from http.server import BaseHTTPRequestHandler
+from typing import TYPE_CHECKING
+
+import pytest
+
+from tests.utils import serve_application
+
+if TYPE_CHECKING:
+    from collections.abc import Iterable
+    from typing import Any
+
+    from sphinx.testing.util import SphinxTestApp
+
+
+class EncodedAnchorsHandler(BaseHTTPRequestHandler):
+    protocol_version = 'HTTP/1.1'
+
+    def _chunk_content(self, content: str, *, max_chunk_size: int) -> Iterable[bytes]:
+        """Split content into chunks of a maximum size."""
+
+        def _encode_chunk(chunk: bytes) -> Iterable[bytes]:
+            """Encode a bytestring into a format suitable for HTTP chunked-transfer."""
+            yield f'{len(chunk):X}'.encode('ascii')
+            yield b'\r\n'
+            yield chunk
+            yield b'\r\n'
+
+        buffer = b''
+        for char in content:
+            buffer += char.encode('utf-8')
+            if len(buffer) >= max_chunk_size:
+                chunk, buffer = buffer[:max_chunk_size], buffer[max_chunk_size:]
+                yield from _encode_chunk(chunk)
+
+        # Flush remaining bytes, if any
+        if buffer:
+            yield from _encode_chunk(buffer)
+
+        # Emit a final empty chunk to close the stream
+        yield from _encode_chunk(b'')
+
+    def _send_chunked(self, content: str) -> bool:
+        """Send content in chunks."""
+        for chunk in self._chunk_content(content, max_chunk_size=20):
+            try:
+                self.wfile.write(chunk)
+            except (BrokenPipeError, ConnectionResetError) as e:
+                self.log_message(str(e))
+                return False
+        return True
+
+    def do_HEAD(self) -> None:
+        """Handle HEAD requests."""
+        print(f'HEAD request for path: {self.path}')
+        if self.path in {'/standard-encoded-anchors', '/various-encoded-chars'}:
+            self.send_response(200, 'OK')
+        else:
+            self.send_response(404, 'Not Found')
+        self.end_headers()
+
+    def do_GET(self) -> None:
+        """Serve test pages with encoded anchors."""
+        if self.path == '/standard-encoded-anchors':
+            self.send_response(200, 'OK')
+            # Note the ID has an encoded forward slash (%2F)
+            content = """
+            <!DOCTYPE html>
+            <html>
+            <head><title>Encoded Anchors Test</title></head>
+            <body>
+                <h1 id="standard-input%2Foutput-stdio">Standard I/O</h1>
+                <h2 id="encoded%2Banchor">Encoded Plus</h2>
+            </body>
+            </html>
+            """
+        elif self.path == '/various-encoded-chars':
+            self.send_response(200, 'OK')
+            content = """
+            <!DOCTYPE html>
+            <html>
+            <head><title>Various Encoded Characters</title></head>
+            <body>
+                <h1 id="encoded%21exclamation">Encoded Exclamation</h1>
+                <h2 id="encoded%23hash">Encoded Hash</h2>
+                <h3 id="encoded%25percent">Encoded Percent</h3>
+                <h4 id="encoded%26ampersand">Encoded Ampersand</h4>
+                <h5 id="encoded%3Fquestion">Encoded Question</h5>
+                <h6 id="encoded%40at">Encoded At</h6>
+            </body>
+            </html>
+            """
+        else:
+            self.send_response(404, 'Not Found')
+            content = 'not found\n'
+        self.send_header('Transfer-Encoding', 'chunked')
+        self.end_headers()
+        self._send_chunked(content)
+
+
+@pytest.mark.sphinx(
+    'linkcheck',
+    testroot='linkcheck-encoded-anchors',
+    freshenv=True,
+)
+def test_encoded_anchors_handling(app: SphinxTestApp, tmp_path: Any) -> None:
+    """Test that linkcheck correctly handles URLs with encoded anchors."""
+    with serve_application(app, EncodedAnchorsHandler) as address:
+        # Create test file with encoded anchor links using the server address
+        (app.srcdir / 'encoded_anchors.rst').write_text(
+            f"""
+Encoded Anchors Test
+====================
+
+Links with encoded anchors:
+
+* `Standard I/O <http://{address}/standard-encoded-anchors#standard-input/output-stdio>`_
+* `Encoded Plus <http://{address}/standard-encoded-anchors#encoded+anchor>`_
+* `Encoded Exclamation <http://{address}/various-encoded-chars#encoded!exclamation>`_
+* `Encoded Hash <http://{address}/various-encoded-chars#encoded#hash>`_
+* `Encoded Percent <http://{address}/various-encoded-chars#encoded%percent>`_
+* `Encoded Ampersand <http://{address}/various-encoded-chars#encoded&ampersand>`_
+* `Encoded Question <http://{address}/various-encoded-chars#encoded?question>`_
+* `Encoded At <http://{address}/various-encoded-chars#encoded@at>`_
+""",
+            encoding='utf-8',
+        )
+
+        app.build()
+
+    # Parse the JSON output to check the results
+    content = (app.outdir / 'output.json').read_text(encoding='utf8')
+    data = [json.loads(record) for record in content.splitlines()]
+
+    # Filter for our encoded anchor URLs
+    encoded_anchor_results = [
+        record
+        for record in data
+        if any(
+            x in record['uri']
+            for x in ['standard-encoded-anchors#', 'various-encoded-chars#']
+        )
+    ]
+
+    # All links with encoded anchors should be working
+    assert all(record['status'] == 'working' for record in encoded_anchor_results)
+
+    # Verify specific links
+    uri_pattern = re.compile(
+        f'http://{re.escape(address)}/standard-encoded-anchors#standard-input/output-stdio'
+    )
+    stdio_link = next(
+        record for record in encoded_anchor_results if uri_pattern.match(record['uri'])
+    )
+    assert stdio_link['status'] == 'working'
+
+    # Check for encoded plus link
+    plus_pattern = re.compile(
+        f'http://{re.escape(address)}/standard-encoded-anchors#encoded\\+anchor'
+    )
+    plus_link = next(
+        record for record in encoded_anchor_results if plus_pattern.match(record['uri'])
+    )
+    assert plus_link['status'] == 'working'