diff --git a/doc/source/whatsnew/v0.25.0.rst b/doc/source/whatsnew/v0.25.0.rst
index fd47ca14dc788..18dcaf1359fe3 100644
--- a/doc/source/whatsnew/v0.25.0.rst
+++ b/doc/source/whatsnew/v0.25.0.rst
@@ -459,6 +459,7 @@ Other API Changes
 - Most Pandas classes had a ``__bytes__`` method, which was used for getting a python2-style bytestring representation of the object. This method has been removed as a part of dropping Python2 (:issue:`26447`)
 - The ``.str``-accessor has been disabled for 1-level :class:`MultiIndex`, use :meth:`MultiIndex.to_flat_index` if necessary (:issue:`23679`)
 - Removed support of gtk package for clipboards (:issue:`26563`)
+- :meth:`ExtensionArray.argsort` places NA values at the end of the sorted array, except :class:`Categorical` (:issue:`21801`)
 
 .. _whatsnew_0250.deprecations:
 
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index c709cd9e9f0b2..088ce9e380174 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -23,6 +23,7 @@
 
 from pandas._typing import ArrayLike
 from pandas.core import ops
+from pandas.core.sorting import nargsort
 
 _not_implemented_message = "{} does not implement {}."
 
@@ -403,13 +404,13 @@ def argsort(self, ascending=True, kind='quicksort', *args, **kwargs):
         """
         # Implementor note: You have two places to override the behavior of
         # argsort.
-        # 1. _values_for_argsort : construct the values passed to np.argsort
+        # 1. _values_for_argsort : construct the values passed to np.argsort.
         # 2. argsort : total control over sorting.
         ascending = nv.validate_argsort_with_ascending(ascending, args, kwargs)
+
         values = self._values_for_argsort()
-        result = np.argsort(values, kind=kind, **kwargs)
-        if not ascending:
-            result = result[::-1]
+        result = nargsort(values, kind=kind, ascending=ascending,
+                          na_position='last')
         return result
 
     def fillna(self, value=None, method=None, limit=None):
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index c079b860bb924..c776031526c48 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -1523,7 +1523,7 @@ def check_for_ordered(self, op):
     def _values_for_argsort(self):
         return self._codes.copy()
 
-    def argsort(self, *args, **kwargs):
+    def argsort(self, ascending=True, kind='quicksort', *args, **kwargs):
         # TODO(PY2): use correct signature
         # We have to do *args, **kwargs to avoid a a py2-only signature
         # issue since np.argsort differs from argsort.
@@ -1567,8 +1567,12 @@ def argsort(self, *args, **kwargs):
         >>> cat.argsort()
         array([3, 0, 1, 2])
         """
-        # Keep the implementation here just for the docstring.
-        return super().argsort(*args, **kwargs)
+        ascending = nv.validate_argsort_with_ascending(ascending, args, kwargs)
+        values = self._values_for_argsort()
+        result = np.argsort(values, kind=kind, **kwargs)
+        if not ascending:
+            result = result[::-1]
+        return result
 
     def sort_values(self, inplace=False, ascending=True, na_position='last'):
         """
diff --git a/pandas/core/arrays/integer.py b/pandas/core/arrays/integer.py
index 29337b7f76131..1dbc8cde595e5 100644
--- a/pandas/core/arrays/integer.py
+++ b/pandas/core/arrays/integer.py
@@ -501,23 +501,6 @@ def value_counts(self, dropna=True):
 
         return Series(array, index=index)
 
-    def _values_for_argsort(self) -> np.ndarray:
-        """Return values for sorting.
-
-        Returns
-        -------
-        ndarray
-            The transformed values should maintain the ordering between values
-            within the array.
-
-        See Also
-        --------
-        ExtensionArray.argsort
-        """
-        data = self._data.copy()
-        data[self._mask] = data.min() - 1
-        return data
-
     @classmethod
     def _create_comparison_method(cls, op):
         def cmp_method(self, other):
diff --git a/pandas/core/arrays/period.py b/pandas/core/arrays/period.py
index a6e282462af68..d22fea29843d4 100644
--- a/pandas/core/arrays/period.py
+++ b/pandas/core/arrays/period.py
@@ -676,7 +676,7 @@ def _check_timedeltalike_freq_compat(self, other):
         _raise_on_incompatible(self, other)
 
     def _values_for_argsort(self):
-        return self._data
+        return self
 
 
 PeriodArray._add_comparison_ops()
diff --git a/pandas/core/sorting.py b/pandas/core/sorting.py
index 21c0c8f747b10..4478166626661 100644
--- a/pandas/core/sorting.py
+++ b/pandas/core/sorting.py
@@ -255,15 +255,17 @@ def nargsort(items, kind='quicksort', ascending=True, na_position='last'):
             sorted_idx = np.roll(sorted_idx, cnt_null)
         return sorted_idx
 
-    with warnings.catch_warnings():
-        # https://github.com/pandas-dev/pandas/issues/25439
-        # can be removed once ExtensionArrays are properly handled by nargsort
-        warnings.filterwarnings(
-            "ignore", category=FutureWarning,
-            message="Converting timezone-aware DatetimeArray to")
-        items = np.asanyarray(items)
-    idx = np.arange(len(items))
     mask = isna(items)
+    if not isinstance(items, np.ndarray):
+        with warnings.catch_warnings():
+            # https://github.com/pandas-dev/pandas/issues/25439
+            # can be removed once ExtensionArrays are properly handled by
+            # nargsort
+            warnings.filterwarnings(
+                "ignore", category=FutureWarning,
+                message="Converting timezone-aware DatetimeArray to")
+            items = np.asanyarray(items)
+    idx = np.arange(len(items))
     non_nans = items[~mask]
     non_nan_idx = idx[~mask]
     nan_idx = np.nonzero(mask)[0]
diff --git a/pandas/tests/extension/base/methods.py b/pandas/tests/extension/base/methods.py
index 1852edaa9e748..50fd4cd292641 100644
--- a/pandas/tests/extension/base/methods.py
+++ b/pandas/tests/extension/base/methods.py
@@ -44,6 +44,11 @@ def test_argsort_missing(self, data_missing_for_sorting):
         expected = pd.Series(np.array([1, -1, 0], dtype=np.int64))
         self.assert_series_equal(result, expected)
 
+    def test_argsort_nan_last(self, data_missing_for_sorting):
+        # GH 21801
+        result = data_missing_for_sorting.argsort()
+        assert result[-1] == 1
+
     @pytest.mark.parametrize('ascending', [True, False])
     def test_sort_values(self, data_for_sorting, ascending):
         ser = pd.Series(data_for_sorting)
diff --git a/pandas/tests/extension/decimal/array.py b/pandas/tests/extension/decimal/array.py
index 1823eeb4d7fc0..ef0e44ec3c01c 100644
--- a/pandas/tests/extension/decimal/array.py
+++ b/pandas/tests/extension/decimal/array.py
@@ -153,6 +153,12 @@ def _reduce(self, name, skipna=True, **kwargs):
                                       "the {} operation".format(name))
         return op(axis=0)
 
+    def _values_for_argsort(self):
+        data = self._data
+        mask = self.isna()
+        data[mask] = decimal.Decimal('Infinity')
+        return data
+
 
 def to_decimal(values, context=None):
     return DecimalArray([decimal.Decimal(x) for x in values], context=context)
diff --git a/pandas/tests/extension/json/array.py b/pandas/tests/extension/json/array.py
index 4b93f0e12e32a..12713a8565228 100644
--- a/pandas/tests/extension/json/array.py
+++ b/pandas/tests/extension/json/array.py
@@ -182,7 +182,10 @@ def _values_for_argsort(self):
         # If all the elemnts of self are the same size P, NumPy will
         # cast them to an (N, P) array, instead of an (N,) array of tuples.
         frozen = [()] + [tuple(x.items()) for x in self]
-        return np.array(frozen, dtype=object)[1:]
+        data = np.array(frozen, dtype=object)[1:]
+        mask = self.isna()
+        data[mask] = np.nan
+        return data
 
 
 def make_data():
diff --git a/pandas/tests/extension/test_categorical.py b/pandas/tests/extension/test_categorical.py
index 4cf9f78e1531d..a169a08fa0d51 100644
--- a/pandas/tests/extension/test_categorical.py
+++ b/pandas/tests/extension/test_categorical.py
@@ -193,6 +193,11 @@ def test_searchsorted(self, data_for_sorting):
         if not data_for_sorting.ordered:
             raise pytest.skip(reason="searchsorted requires ordered data.")
 
+    def test_argsort_nan_last(self, data_missing_for_sorting):
+        # GH 21801
+        # TODO: Categorical.argsort places NA values at the end
+        pass
+
 
 class TestCasting(base.BaseCastingTests):
     pass
diff --git a/pandas/tests/test_sorting.py b/pandas/tests/test_sorting.py
index 75fa37eb9af09..fb23c8822a000 100644
--- a/pandas/tests/test_sorting.py
+++ b/pandas/tests/test_sorting.py
@@ -7,7 +7,8 @@
 import pytest
 
 from pandas import (
-    DataFrame, MultiIndex, Series, array, concat, merge, to_datetime)
+    Categorical, DataFrame, MultiIndex, Series, array, concat, merge,
+    to_datetime)
 from pandas.core import common as com
 from pandas.core.sorting import (
     decons_group_index, get_group_index, is_int64_overflow_possible,
@@ -188,6 +189,14 @@ def test_nargsort_datetimearray_warning(self):
         with tm.assert_produces_warning(None):
             nargsort(data)
 
+    @pytest.mark.parametrize('data', [
+        Categorical(['a', 'c', 'a', 'b']),
+        to_datetime([0, 2, 0, 1]).tz_localize('Europe/Brussels')])
+    def test_nargsort_extension_array(self, data):
+        result = nargsort(data)
+        expected = np.array([0, 2, 3, 1], dtype=np.intp)
+        tm.assert_numpy_array_equal(result, expected)
+
 
 class TestMerge: