diff --git a/pandas/core/arrays/string_.py b/pandas/core/arrays/string_.py
index 01619dab7ce45..de129df2575d3 100644
--- a/pandas/core/arrays/string_.py
+++ b/pandas/core/arrays/string_.py
@@ -846,7 +846,7 @@ def _reduce(
             else:
                 return nanops.nanall(self._ndarray, skipna=skipna)
 
-        if name in ["min", "max", "sum"]:
+        if name in ["min", "max", "argmin", "argmax", "sum"]:
             result = getattr(self, name)(skipna=skipna, axis=axis, **kwargs)
             if keepdims:
                 return self._from_sequence([result], dtype=self.dtype)
diff --git a/pandas/tests/frame/test_reductions.py b/pandas/tests/frame/test_reductions.py
index 30d02f9b5463d..fde4dfeed9c55 100644
--- a/pandas/tests/frame/test_reductions.py
+++ b/pandas/tests/frame/test_reductions.py
@@ -6,8 +6,6 @@
 import numpy as np
 import pytest
 
-from pandas._config import using_string_dtype
-
 from pandas.compat import (
     IS64,
     is_platform_windows,
@@ -1081,7 +1079,6 @@ def test_idxmin_empty(self, index, skipna, axis):
         expected = Series(dtype=index.dtype)
         tm.assert_series_equal(result, expected)
 
-    @pytest.mark.xfail(using_string_dtype(), reason="TODO(infer_string)", strict=False)
     @pytest.mark.parametrize("numeric_only", [True, False])
     def test_idxmin_numeric_only(self, numeric_only):
         df = DataFrame({"a": [2, 3, 1], "b": [2, 1, 1], "c": list("xyx")})
@@ -1098,7 +1095,6 @@ def test_idxmin_axis_2(self, float_frame):
         with pytest.raises(ValueError, match=msg):
             frame.idxmin(axis=2)
 
-    @pytest.mark.xfail(using_string_dtype(), reason="TODO(infer_string)", strict=False)
     @pytest.mark.parametrize("axis", [0, 1])
     def test_idxmax(self, float_frame, int_frame, skipna, axis):
         frame = float_frame
@@ -1132,7 +1128,6 @@ def test_idxmax_empty(self, index, skipna, axis):
         expected = Series(dtype=index.dtype)
         tm.assert_series_equal(result, expected)
 
-    @pytest.mark.xfail(using_string_dtype(), reason="TODO(infer_string)", strict=False)
     @pytest.mark.parametrize("numeric_only", [True, False])
     def test_idxmax_numeric_only(self, numeric_only):
         df = DataFrame({"a": [2, 3, 1], "b": [2, 1, 1], "c": list("xyx")})
diff --git a/pandas/tests/reductions/test_reductions.py b/pandas/tests/reductions/test_reductions.py
index 8153ba66d632b..476978aeab15a 100644
--- a/pandas/tests/reductions/test_reductions.py
+++ b/pandas/tests/reductions/test_reductions.py
@@ -7,10 +7,6 @@
 import numpy as np
 import pytest
 
-from pandas._config import using_string_dtype
-
-from pandas.compat import HAS_PYARROW
-
 import pandas as pd
 from pandas import (
     Categorical,
@@ -1206,10 +1202,6 @@ def test_idxminmax_object_dtype(self, using_infer_string):
             with pytest.raises(TypeError, match=msg):
                 ser3.idxmin(skipna=False)
 
-    # TODO(infer_string) implement argmin/max for python string dtype
-    @pytest.mark.xfail(
-        using_string_dtype() and not HAS_PYARROW, reason="TODO(infer_string)"
-    )
     def test_idxminmax_object_frame(self):
         # GH#4279
         df = DataFrame([["zimm", 2.5], ["biff", 1.0], ["bid", 12.0]])