diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 607a14c696578..eb9e31be0772e 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -302,8 +302,9 @@ Sparse
 
 ExtensionArray
 ^^^^^^^^^^^^^^
+
 - Bug in :meth:`DataFrame.where` when ``other`` is a :class:`Series` with ExtensionArray dtype (:issue:`38729`)
--
+- Fixed bug where :meth:`Series.idxmax`, :meth:`Series.idxmin` and ``argmax/min`` fail when the underlying data is :class:`ExtensionArray` (:issue:`32749`, :issue:`33719`, :issue:`36566`)
 -
 
 Other
diff --git a/pandas/core/base.py b/pandas/core/base.py
index 54dec90c08aa2..afc22a8446dce 100644
--- a/pandas/core/base.py
+++ b/pandas/core/base.py
@@ -715,9 +715,17 @@ def argmax(self, axis=None, skipna: bool = True, *args, **kwargs) -> int:
         the minimum cereal calories is the first element,
         since series is zero-indexed.
         """
+        delegate = self._values
         nv.validate_minmax_axis(axis)
-        nv.validate_argmax_with_skipna(skipna, args, kwargs)
-        return nanops.nanargmax(self._values, skipna=skipna)
+        skipna = nv.validate_argmax_with_skipna(skipna, args, kwargs)
+
+        if isinstance(delegate, ExtensionArray):
+            if not skipna and delegate.isna().any():
+                return -1
+            else:
+                return delegate.argmax()
+        else:
+            return nanops.nanargmax(delegate, skipna=skipna)
 
     def min(self, axis=None, skipna: bool = True, *args, **kwargs):
         """
@@ -765,9 +773,17 @@ def min(self, axis=None, skipna: bool = True, *args, **kwargs):
 
     @doc(argmax, op="min", oppose="max", value="smallest")
     def argmin(self, axis=None, skipna=True, *args, **kwargs) -> int:
+        delegate = self._values
         nv.validate_minmax_axis(axis)
-        nv.validate_argmax_with_skipna(skipna, args, kwargs)
-        return nanops.nanargmin(self._values, skipna=skipna)
+        skipna = nv.validate_argmin_with_skipna(skipna, args, kwargs)
+
+        if isinstance(delegate, ExtensionArray):
+            if not skipna and delegate.isna().any():
+                return -1
+            else:
+                return delegate.argmin()
+        else:
+            return nanops.nanargmin(delegate, skipna=skipna)
 
     def tolist(self):
         """
diff --git a/pandas/core/series.py b/pandas/core/series.py
index cb753e887b0f8..7f2039c998f53 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -2076,8 +2076,7 @@ def idxmin(self, axis=0, skipna=True, *args, **kwargs):
         >>> s.idxmin(skipna=False)
         nan
         """
-        skipna = nv.validate_argmin_with_skipna(skipna, args, kwargs)
-        i = nanops.nanargmin(self._values, skipna=skipna)
+        i = self.argmin(None, skipna=skipna)
         if i == -1:
             return np.nan
         return self.index[i]
@@ -2147,8 +2146,7 @@ def idxmax(self, axis=0, skipna=True, *args, **kwargs):
         >>> s.idxmax(skipna=False)
         nan
         """
-        skipna = nv.validate_argmax_with_skipna(skipna, args, kwargs)
-        i = nanops.nanargmax(self._values, skipna=skipna)
+        i = self.argmax(None, skipna=skipna)
         if i == -1:
             return np.nan
         return self.index[i]
diff --git a/pandas/tests/extension/base/methods.py b/pandas/tests/extension/base/methods.py
index 1cc03d4f4f2bd..472e783c977f0 100644
--- a/pandas/tests/extension/base/methods.py
+++ b/pandas/tests/extension/base/methods.py
@@ -107,6 +107,27 @@ def test_argmin_argmax_all_na(self, method, data, na_value):
         with pytest.raises(ValueError, match=err_msg):
             getattr(data_na, method)()
 
+    @pytest.mark.parametrize(
+        "op_name, skipna, expected",
+        [
+            ("idxmax", True, 0),
+            ("idxmin", True, 2),
+            ("argmax", True, 0),
+            ("argmin", True, 2),
+            ("idxmax", False, np.nan),
+            ("idxmin", False, np.nan),
+            ("argmax", False, -1),
+            ("argmin", False, -1),
+        ],
+    )
+    def test_argreduce_series(
+        self, data_missing_for_sorting, op_name, skipna, expected
+    ):
+        # data_missing_for_sorting -> [B, NA, A] with A < B and NA missing.
+        ser = pd.Series(data_missing_for_sorting)
+        result = getattr(ser, op_name)(skipna=skipna)
+        tm.assert_almost_equal(result, expected)
+
     @pytest.mark.parametrize(
         "na_position, expected",
         [