diff --git a/tests/test_completion/test_completion_bash_version.py b/tests/test_completion/test_completion_bash_version.py
new file mode 100644
index 0000000000..1d76a24680
--- /dev/null
+++ b/tests/test_completion/test_completion_bash_version.py
@@ -0,0 +1,54 @@
+import shutil
+import subprocess
+from unittest import mock
+
+import pytest
+from typer import _click
+from typer._completion_classes import BashComplete
+
+
+def _check_version_with(bash_version: str) -> bool:
+    """Run BashComplete._check_version against a faked Bash and report
+    whether it warned that completion is unsupported.
+    """
+    completed = subprocess.CompletedProcess(
+        args=["bash", "--norc", "-c", 'echo "${BASH_VERSION}"'],
+        returncode=0,
+        stdout=f"{bash_version}\n".encode(),
+    )
+    with (
+        mock.patch.object(shutil, "which", return_value="/usr/bin/bash"),
+        mock.patch.object(subprocess, "run", return_value=completed),
+        mock.patch.object(_click.utils, "echo") as echo,
+    ):
+        BashComplete._check_version()
+    return echo.called
+
+
+@pytest.mark.parametrize(
+    "bash_version",
+    ["3.2.57", "4.0.0", "4.3.9", "4.3.99"],
+)
+def test_bash_too_old_is_reported_unsupported(bash_version: str):
+    assert _check_version_with(bash_version) is True
+
+
+@pytest.mark.parametrize(
+    "bash_version",
+    ["4.4.0", "4.5.0", "4.9.0", "5.0.0", "5.3.0"],
+)
+def test_bash_new_enough_is_supported(bash_version: str):
+    assert _check_version_with(bash_version) is False
+
+
+@pytest.mark.parametrize(
+    "bash_version",
+    ["4.10.0", "4.11.2", "5.10.0", "10.0.0", "12.3.4"],
+)
+def test_bash_two_digit_version_components_are_supported(bash_version: str):
+    """Version components must be compared numerically.
+
+    "10" sorts before "4" as a string, so a lexicographic comparison wrongly
+    reports these versions as older than 4.4.
+    """
+    assert _check_version_with(bash_version) is False
diff --git a/typer/_completion_classes.py b/typer/_completion_classes.py
index cfae02c9bf..4af5804f53 100644
--- a/typer/_completion_classes.py
+++ b/typer/_completion_classes.py
@@ -77,9 +77,9 @@ def _check_version() -> None:
             match = re.search(r"^(\d+)\.(\d+)\.\d+", output.stdout.decode())
 
         if match is not None:
-            major, minor = match.groups()
+            major, minor = (int(part) for part in match.groups())
 
-            if major < "4" or major == "4" and minor < "4":
+            if major < 4 or (major == 4 and minor < 4):
                 _click.utils.echo(
                     "Shell completion is not supported for Bash versions older than 4.4.",
                     err=True,