diff --git a/doc/source/whatsnew/v2.0.0.rst b/doc/source/whatsnew/v2.0.0.rst
index b53595621af11..554c972b56b34 100644
--- a/doc/source/whatsnew/v2.0.0.rst
+++ b/doc/source/whatsnew/v2.0.0.rst
@@ -946,6 +946,7 @@ Conversion
 Strings
 ^^^^^^^
 - Bug in :func:`pandas.api.dtypes.is_string_dtype` that would not return ``True`` for :class:`StringDtype` (:issue:`15585`)
+- Bug in converting string dtypes to "datetime64[ns]" or "timedelta64[ns]" incorrectly raising ``TypeError`` (:issue:`36153`)
 -
 
 Interval
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index 422b9effeface..4497583f60d71 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -55,9 +55,11 @@
 
 from pandas.core.dtypes.cast import maybe_cast_to_extension_array
 from pandas.core.dtypes.common import (
+    is_datetime64_dtype,
     is_dtype_equal,
     is_list_like,
     is_scalar,
+    is_timedelta64_dtype,
     pandas_dtype,
 )
 from pandas.core.dtypes.dtypes import ExtensionDtype
@@ -580,6 +582,16 @@ def astype(self, dtype: AstypeArg, copy: bool = True) -> ArrayLike:
             cls = dtype.construct_array_type()
             return cls._from_sequence(self, dtype=dtype, copy=copy)
 
+        elif is_datetime64_dtype(dtype):
+            from pandas.core.arrays import DatetimeArray
+
+            return DatetimeArray._from_sequence(self, dtype=dtype, copy=copy)
+
+        elif is_timedelta64_dtype(dtype):
+            from pandas.core.arrays import TimedeltaArray
+
+            return TimedeltaArray._from_sequence(self, dtype=dtype, copy=copy)
+
         return np.array(self, dtype=dtype, copy=copy)
 
     def isna(self) -> np.ndarray | ExtensionArraySupportsAnyAll:
diff --git a/pandas/tests/arrays/string_/test_string.py b/pandas/tests/arrays/string_/test_string.py
index 8d8d9ce20cefd..e1ea001819b1c 100644
--- a/pandas/tests/arrays/string_/test_string.py
+++ b/pandas/tests/arrays/string_/test_string.py
@@ -73,17 +73,7 @@ def test_setitem_with_scalar_string(dtype):
     tm.assert_extension_array_equal(arr, expected)
 
 
-def test_astype_roundtrip(dtype, request):
-    if dtype.storage == "pyarrow":
-        reason = "ValueError: Could not convert object to NumPy datetime"
-        mark = pytest.mark.xfail(reason=reason, raises=ValueError)
-        request.node.add_marker(mark)
-    else:
-        mark = pytest.mark.xfail(
-            reason="GH#36153 casting from StringArray to dt64 fails", raises=ValueError
-        )
-        request.node.add_marker(mark)
-
+def test_astype_roundtrip(dtype):
     ser = pd.Series(pd.date_range("2000", periods=12))
     ser[0] = None
 
diff --git a/pandas/tests/series/methods/test_astype.py b/pandas/tests/series/methods/test_astype.py
index 9cc49199166b8..024cdf9300157 100644
--- a/pandas/tests/series/methods/test_astype.py
+++ b/pandas/tests/series/methods/test_astype.py
@@ -454,9 +454,7 @@ class TestAstypeString:
     def test_astype_string_to_extension_dtype_roundtrip(
         self, data, dtype, request, nullable_string_dtype
     ):
-        if dtype == "boolean" or (
-            dtype in ("datetime64[ns]", "timedelta64[ns]") and NaT in data
-        ):
+        if dtype == "boolean" or (dtype == "timedelta64[ns]" and NaT in data):
             mark = pytest.mark.xfail(
                 reason="TODO StringArray.astype() with missing values #GH40566"
             )