diff --git a/doc/source/whatsnew/v2.0.0.rst b/doc/source/whatsnew/v2.0.0.rst
index 75aec514031b4..14ae995d2fd57 100644
--- a/doc/source/whatsnew/v2.0.0.rst
+++ b/doc/source/whatsnew/v2.0.0.rst
@@ -308,6 +308,7 @@ Other enhancements
 - :meth:`Series.dropna` and :meth:`DataFrame.dropna` has gained ``ignore_index`` keyword to reset index (:issue:`31725`)
 - Improved error message in :func:`to_datetime` for non-ISO8601 formats, informing users about the position of the first error (:issue:`50361`)
 - Improved error message when trying to align :class:`DataFrame` objects (for example, in :func:`DataFrame.compare`) to clarify that "identically labelled" refers to both index and columns (:issue:`50083`)
+- Added support for :meth:`Index.min` and :meth:`Index.max` for pyarrow string dtypes (:issue:`51397`)
 - Added :meth:`DatetimeIndex.as_unit` and :meth:`TimedeltaIndex.as_unit` to convert to different resolutions; supported resolutions are "s", "ms", "us", and "ns" (:issue:`50616`)
 - Added :meth:`Series.dt.unit` and :meth:`Series.dt.as_unit` to convert to different resolutions; supported resolutions are "s", "ms", "us", and "ns" (:issue:`51223`)
 - Added new argument ``dtype`` to :func:`read_sql` to be consistent with :func:`read_sql_query` (:issue:`50797`)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 6031bdc62c38a..31e34379fd373 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -6928,8 +6928,7 @@ def min(self, axis=None, skipna: bool = True, *args, **kwargs):
                 return self._na_value
 
         if not self._is_multi and not isinstance(self._values, np.ndarray):
-            # "ExtensionArray" has no attribute "min"
-            return self._values.min(skipna=skipna)  # type: ignore[attr-defined]
+            return self._values._reduce(name="min", skipna=skipna)
 
         return super().min(skipna=skipna)
 
@@ -6954,8 +6953,7 @@ def max(self, axis=None, skipna: bool = True, *args, **kwargs):
                 return self._na_value
 
         if not self._is_multi and not isinstance(self._values, np.ndarray):
-            # "ExtensionArray" has no attribute "max"
-            return self._values.max(skipna=skipna)  # type: ignore[attr-defined]
+            return self._values._reduce(name="max", skipna=skipna)
 
         return super().max(skipna=skipna)
 
diff --git a/pandas/tests/indexes/test_numpy_compat.py b/pandas/tests/indexes/test_numpy_compat.py
index 788d03d5b7212..07ebe6fa04beb 100644
--- a/pandas/tests/indexes/test_numpy_compat.py
+++ b/pandas/tests/indexes/test_numpy_compat.py
@@ -158,10 +158,6 @@ def test_numpy_ufuncs_reductions(index, func, request):
     if len(index) == 0:
         return
 
-    if repr(index.dtype) == "string[pyarrow]":
-        mark = pytest.mark.xfail(reason="ArrowStringArray has no min/max")
-        request.node.add_marker(mark)
-
     if isinstance(index, CategoricalIndex) and index.dtype.ordered is False:
         with pytest.raises(TypeError, match="is not ordered for"):
             func.reduce(index)