diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index fcec562279248..42d85935efe95 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -528,8 +528,12 @@ def _argmin_max(self, skipna: bool, method: str) -> int:
                 f"arg{method} only implemented for pyarrow version >= 6.0"
             )
 
-        value = getattr(pc, method)(self._data, skip_nulls=skipna)
-        return pc.index(self._data, value).as_py()
+        data = self._data
+        if pa.types.is_duration(data.type):
+            data = data.cast(pa.int64())
+
+        value = getattr(pc, method)(data, skip_nulls=skipna)
+        return pc.index(data, value).as_py()
 
     def argmin(self, skipna: bool = True) -> int:
         return self._argmin_max(skipna, "min")
diff --git a/pandas/tests/extension/test_arrow.py b/pandas/tests/extension/test_arrow.py
index f7517e1a3ab7f..c798a3d91f816 100644
--- a/pandas/tests/extension/test_arrow.py
+++ b/pandas/tests/extension/test_arrow.py
@@ -869,13 +869,6 @@ def test_argmin_argmax(
                     reason=f"{pa_dtype} only has 2 unique possible values",
                 )
             )
-        elif pa.types.is_duration(pa_dtype):
-            request.node.add_marker(
-                pytest.mark.xfail(
-                    raises=pa.ArrowNotImplementedError,
-                    reason=f"min_max not supported in pyarrow for {pa_dtype}",
-                )
-            )
         super().test_argmin_argmax(data_for_sorting, data_missing_for_sorting, na_value)
 
     @pytest.mark.parametrize(
@@ -894,7 +887,6 @@ def test_argmin_argmax(
     def test_argreduce_series(
         self, data_missing_for_sorting, op_name, skipna, expected, request
     ):
-        pa_dtype = data_missing_for_sorting.dtype.pyarrow_dtype
         if pa_version_under6p0 and skipna:
             request.node.add_marker(
                 pytest.mark.xfail(
@@ -902,13 +894,6 @@ def test_argreduce_series(
                     reason="min_max not supported in pyarrow",
                 )
             )
-        elif not pa_version_under6p0 and pa.types.is_duration(pa_dtype) and skipna:
-            request.node.add_marker(
-                pytest.mark.xfail(
-                    raises=pa.ArrowNotImplementedError,
-                    reason=f"min_max not supported in pyarrow for {pa_dtype}",
-                )
-            )
         super().test_argreduce_series(
             data_missing_for_sorting, op_name, skipna, expected
         )