diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index 85351945cf29c..636a22fcffe3d 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -1596,6 +1596,15 @@ 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:
@@ -1606,7 +1615,7 @@ 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 9b3245183bd57..2eef240af53f8 100644
--- a/pandas/core/arrays/string_arrow.py
+++ b/pandas/core/arrays/string_arrow.py
@@ -445,6 +445,17 @@ def _str_rstrip(self, to_strip=None):
             result = pc.utf8_rtrim(self._pa_array, characters=to_strip)
         return type(self)(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 _convert_int_dtype(self, result):
         return Int64Dtype().__from_arrow__(result)
 
diff --git a/pandas/tests/frame/test_reductions.py b/pandas/tests/frame/test_reductions.py
index 4c83c810e8cec..0b501903ad71f 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"