diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index 12fe9b30f3f52..473f2c1ff2222 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -1627,6 +1627,15 @@ def _reduce(
         ------
         TypeError : subclass does not define reductions
         """
+        result = self._reduce_calc(name, skipna=skipna, keepdims=keepdims, **kwargs)
+        if isinstance(result, pa.Array):
+            return type(self)(result)
+        else:
+            return result
+
+    def _reduce_calc(
+        self, name: str, *, skipna: bool = True, keepdims: bool = False, **kwargs
+    ):
         pa_result = self._reduce_pyarrow(name, skipna=skipna, **kwargs)
 
         if keepdims:
@@ -1637,7 +1646,7 @@ def _reduce(
                     [pa_result],
                     type=to_pyarrow_type(infer_dtype_from_scalar(pa_result)[0]),
                 )
-            return type(self)(result)
+            return result
 
         if pc.is_null(pa_result).as_py():
             return self.dtype.na_value
diff --git a/pandas/core/arrays/string_arrow.py b/pandas/core/arrays/string_arrow.py
index 24b99b5d4852e..2a10e87981bc3 100644
--- a/pandas/core/arrays/string_arrow.py
+++ b/pandas/core/arrays/string_arrow.py
@@ -502,6 +502,17 @@ def _str_find(self, sub: str, start: int = 0, end: int | None = None):
     def _convert_int_dtype(self, result):
         return Int64Dtype().__from_arrow__(result)
 
+    def _reduce(
+        self, name: str, *, skipna: bool = True, keepdims: bool = False, **kwargs
+    ):
+        result = self._reduce_calc(name, skipna=skipna, keepdims=keepdims, **kwargs)
+        if name in ("argmin", "argmax") and isinstance(result, pa.Array):
+            return self._convert_int_dtype(result)
+        elif isinstance(result, pa.Array):
+            return type(self)(result)
+        else:
+            return result
+
     def _rank(
         self,
         *,
diff --git a/pandas/tests/frame/test_reductions.py b/pandas/tests/frame/test_reductions.py
index 66f813c80ed16..e3c8958f39018 100644
--- a/pandas/tests/frame/test_reductions.py
+++ b/pandas/tests/frame/test_reductions.py
@@ -1073,6 +1073,15 @@ def test_idxmax_arrow_types(self):
         expected = Series([2, 1], index=["a", "b"])
         tm.assert_series_equal(result, expected)
 
+        df = DataFrame({"a": ["b", "c", "a"]}, dtype="string[pyarrow]")
+        result = df.idxmax(numeric_only=False)
+        expected = Series([1], index=["a"])
+        tm.assert_series_equal(result, expected)
+
+        result = df.idxmin(numeric_only=False)
+        expected = Series([2], index=["a"])
+        tm.assert_series_equal(result, expected)
+
     def test_idxmax_axis_2(self, float_frame):
         frame = float_frame
         msg = "No axis named 2 for object type DataFrame"