diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index cd1cb0b64f74a..0d7bb75b060e2 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -499,6 +499,7 @@ Strings
 ^^^^^^^
 
 - Bug in the :meth:`~Series.astype` method when converting "string" dtype data to nullable integer dtype (:issue:`32450`).
+- Fixed issue where taking ``min`` or ``max`` of a ``StringArray`` or ``Series`` with ``StringDtype`` type would raise. (:issue:`31746`)
 - Bug in :meth:`Series.str.cat` returning ``NaN`` output when other had :class:`Index` type (:issue:`33425`)
 
 
diff --git a/pandas/compat/numpy/function.py b/pandas/compat/numpy/function.py
index 260cc69187d38..e988f5d97f7ee 100644
--- a/pandas/compat/numpy/function.py
+++ b/pandas/compat/numpy/function.py
@@ -218,7 +218,7 @@ def validate_cum_func_with_skipna(skipna, args, kwargs, name):
 LOGICAL_FUNC_DEFAULTS = dict(out=None, keepdims=False)
 validate_logical_func = CompatValidator(LOGICAL_FUNC_DEFAULTS, method="kwargs")
 
-MINMAX_DEFAULTS = dict(out=None, keepdims=False)
+MINMAX_DEFAULTS = dict(axis=None, out=None, keepdims=False)
 validate_min = CompatValidator(
     MINMAX_DEFAULTS, fname="min", method="both", max_fname_arg_count=1
 )
diff --git a/pandas/core/arrays/numpy_.py b/pandas/core/arrays/numpy_.py
index 3058e1d6073f3..e9950e0edaffb 100644
--- a/pandas/core/arrays/numpy_.py
+++ b/pandas/core/arrays/numpy_.py
@@ -5,6 +5,7 @@
 from numpy.lib.mixins import NDArrayOperatorsMixin
 
 from pandas._libs import lib
+from pandas._typing import Scalar
 from pandas.compat.numpy import function as nv
 from pandas.util._decorators import doc
 from pandas.util._validators import validate_fillna_kwargs
@@ -17,6 +18,7 @@
 from pandas import compat
 from pandas.core import nanops
 from pandas.core.algorithms import searchsorted, take, unique
+from pandas.core.array_algos import masked_reductions
 from pandas.core.arrays.base import ExtensionArray, ExtensionOpsMixin
 from pandas.core.construction import extract_array
 from pandas.core.indexers import check_array_indexer
@@ -349,13 +351,19 @@ def all(self, axis=None, out=None, keepdims=False, skipna=True):
         nv.validate_all((), dict(out=out, keepdims=keepdims))
         return nanops.nanall(self._ndarray, axis=axis, skipna=skipna)
 
-    def min(self, axis=None, out=None, keepdims=False, skipna=True):
-        nv.validate_min((), dict(out=out, keepdims=keepdims))
-        return nanops.nanmin(self._ndarray, axis=axis, skipna=skipna)
+    def min(self, skipna: bool = True, **kwargs) -> Scalar:
+        nv.validate_min((), kwargs)
+        result = masked_reductions.min(
+            values=self.to_numpy(), mask=self.isna(), skipna=skipna
+        )
+        return result
 
-    def max(self, axis=None, out=None, keepdims=False, skipna=True):
-        nv.validate_max((), dict(out=out, keepdims=keepdims))
-        return nanops.nanmax(self._ndarray, axis=axis, skipna=skipna)
+    def max(self, skipna: bool = True, **kwargs) -> Scalar:
+        nv.validate_max((), kwargs)
+        result = masked_reductions.max(
+            values=self.to_numpy(), mask=self.isna(), skipna=skipna
+        )
+        return result
 
     def sum(
         self,
diff --git a/pandas/core/arrays/string_.py b/pandas/core/arrays/string_.py
index dbca8e74f5e1b..51bbe182a002b 100644
--- a/pandas/core/arrays/string_.py
+++ b/pandas/core/arrays/string_.py
@@ -282,6 +282,9 @@ def astype(self, dtype, copy=True):
         return super().astype(dtype, copy)
 
     def _reduce(self, name, skipna=True, **kwargs):
+        if name in ["min", "max"]:
+            return getattr(self, name)(skipna=skipna)
+
         raise TypeError(f"Cannot perform reduction '{name}' with string dtype")
 
     def value_counts(self, dropna=False):
diff --git a/pandas/tests/arrays/string_/test_string.py b/pandas/tests/arrays/string_/test_string.py
index fe770eed84b62..eb89798a1ad96 100644
--- a/pandas/tests/arrays/string_/test_string.py
+++ b/pandas/tests/arrays/string_/test_string.py
@@ -230,6 +230,32 @@ def test_reduce(skipna):
     assert result == "abc"
 
 
+@pytest.mark.parametrize("method", ["min", "max"])
+@pytest.mark.parametrize("skipna", [True, False])
+def test_min_max(method, skipna):
+    arr = pd.Series(["a", "b", "c", None], dtype="string")
+    result = getattr(arr, method)(skipna=skipna)
+    if skipna:
+        expected = "a" if method == "min" else "c"
+        assert result == expected
+    else:
+        assert result is pd.NA
+
+
+@pytest.mark.parametrize("method", ["min", "max"])
+@pytest.mark.parametrize(
+    "arr",
+    [
+        pd.Series(["a", "b", "c", None], dtype="string"),
+        pd.array(["a", "b", "c", None], dtype="string"),
+    ],
+)
+def test_min_max_numpy(method, arr):
+    result = getattr(np, method)(arr)
+    expected = "a" if method == "min" else "c"
+    assert result == expected
+
+
 @pytest.mark.parametrize("skipna", [True, False])
 @pytest.mark.xfail(reason="Not implemented StringArray.sum")
 def test_reduce_missing(skipna):
diff --git a/pandas/tests/extension/test_string.py b/pandas/tests/extension/test_string.py
index 86aed671f1b88..27a157d2127f6 100644
--- a/pandas/tests/extension/test_string.py
+++ b/pandas/tests/extension/test_string.py
@@ -77,7 +77,16 @@ class TestMissing(base.BaseMissingTests):
 
 
 class TestNoReduce(base.BaseNoReduceTests):
-    pass
+    @pytest.mark.parametrize("skipna", [True, False])
+    def test_reduce_series_numeric(self, data, all_numeric_reductions, skipna):
+        op_name = all_numeric_reductions
+
+        if op_name in ["min", "max"]:
+            return None
+
+        s = pd.Series(data)
+        with pytest.raises(TypeError):
+            getattr(s, op_name)(skipna=skipna)
 
 
 class TestMethods(base.BaseMethodsTests):