diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index c9347b88f2072..6aff4da13b6e6 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -784,6 +784,9 @@ Other
 - Passing an array with 2 or more dimensions to the :class:`Series` constructor now raises the more specific ``ValueError`` rather than a bare ``Exception`` (:issue:`35744`)
 - Bug in ``dir`` where ``dir(obj)`` wouldn't show attributes defined on the instance for pandas objects (:issue:`37173`)
 - Bug in :meth:`RangeIndex.difference` returning :class:`Int64Index` in some cases where it should return :class:`RangeIndex` (:issue:`38028`)
+- Fixed bug in :func:`assert_series_equal` when comparing a datetime-like array with an equivalent non extension dtype array (:issue:`37609`)
+
+
 
 .. ---------------------------------------------------------------------------
 
diff --git a/pandas/_testing.py b/pandas/_testing.py
index bfff4301c2220..469f5e1bed6ba 100644
--- a/pandas/_testing.py
+++ b/pandas/_testing.py
@@ -1456,7 +1456,16 @@ def assert_series_equal(
             check_dtype=check_dtype,
             index_values=np.asarray(left.index),
         )
-    elif needs_i8_conversion(left.dtype) or needs_i8_conversion(right.dtype):
+    elif is_extension_array_dtype_and_needs_i8_conversion(
+        left.dtype, right.dtype
+    ) or is_extension_array_dtype_and_needs_i8_conversion(right.dtype, left.dtype):
+        assert_extension_array_equal(
+            left._values,
+            right._values,
+            check_dtype=check_dtype,
+            index_values=np.asarray(left.index),
+        )
+    elif needs_i8_conversion(left.dtype) and needs_i8_conversion(right.dtype):
         # DatetimeArray or TimedeltaArray
         assert_extension_array_equal(
             left._values,
@@ -1866,6 +1875,20 @@ def assert_copy(iter1, iter2, **eql_kwargs):
         assert elem1 is not elem2, msg
 
 
+def is_extension_array_dtype_and_needs_i8_conversion(left_dtype, right_dtype) -> bool:
+    """
+    Checks that we have the combination of an ExtensionArraydtype and
+    a dtype that should be converted to int64
+
+    Returns
+    -------
+    bool
+
+    Related to issue #37609
+    """
+    return is_extension_array_dtype(left_dtype) and needs_i8_conversion(right_dtype)
+
+
 def getCols(k):
     return string.ascii_uppercase[:k]
 
diff --git a/pandas/tests/util/test_assert_frame_equal.py b/pandas/tests/util/test_assert_frame_equal.py
index 40d2763a13489..8034ace479a62 100644
--- a/pandas/tests/util/test_assert_frame_equal.py
+++ b/pandas/tests/util/test_assert_frame_equal.py
@@ -272,6 +272,20 @@ def test_assert_frame_equal_ignore_extension_dtype_mismatch(right_dtype):
     tm.assert_frame_equal(left, right, check_dtype=False)
 
 
+@pytest.mark.parametrize(
+    "dtype",
+    [
+        ("timedelta64[ns]"),
+        ("datetime64[ns, UTC]"),
+        ("Period[D]"),
+    ],
+)
+def test_assert_frame_equal_datetime_like_dtype_mismatch(dtype):
+    df1 = DataFrame({"a": []}, dtype=dtype)
+    df2 = DataFrame({"a": []})
+    tm.assert_frame_equal(df1, df2, check_dtype=False)
+
+
 def test_allows_duplicate_labels():
     left = DataFrame()
     right = DataFrame().set_flags(allows_duplicate_labels=False)