diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 9c424f70b1ee0..8dba9ff693d31 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -560,6 +560,7 @@ Datetimelike
 - Bug in :meth:`DatetimeIndex.intersection` losing ``freq`` and timezone in some cases (:issue:`33604`)
 - Bug in :class:`DatetimeIndex` addition and subtraction with some types of :class:`DateOffset` objects incorrectly retaining an invalid ``freq`` attribute (:issue:`33779`)
 - Bug in :class:`DatetimeIndex` where setting the ``freq`` attribute on an index could silently change the ``freq`` attribute on another index viewing the same data (:issue:`33552`)
+- :meth:`DataFrame.min`/:meth:`DataFrame.max` not returning consistent result with :meth:`Series.min`/:meth:`Series.max` when called on objects initialized with empty :func:`pd.to_datetime`
 - Bug in :meth:`DatetimeIndex.intersection` and :meth:`TimedeltaIndex.intersection` with results not having the correct ``name`` attribute (:issue:`33904`)
 - Bug in :meth:`DatetimeArray.__setitem__`, :meth:`TimedeltaArray.__setitem__`, :meth:`PeriodArray.__setitem__` incorrectly allowing values with ``int64`` dtype to be silently cast (:issue:`33717`)
 
diff --git a/pandas/core/nanops.py b/pandas/core/nanops.py
index bab9df0b70598..741ef64dbbf3b 100644
--- a/pandas/core/nanops.py
+++ b/pandas/core/nanops.py
@@ -384,8 +384,12 @@ def _na_for_min_count(
     else:
         assert axis is not None  # assertion to make mypy happy
         result_shape = values.shape[:axis] + values.shape[axis + 1 :]
-        result = np.empty(result_shape, dtype=values.dtype)
-        result.fill(fill_value)
+        # calling np.full with dtype parameter throws an ValueError when called
+        # with dtype=np.datetime64 and and fill_value=pd.NaT
+        try:
+            result = np.full(result_shape, fill_value, dtype=values.dtype)
+        except ValueError:
+            result = np.full(result_shape, fill_value)
         return result
 
 
diff --git a/pandas/tests/frame/test_analytics.py b/pandas/tests/frame/test_analytics.py
index f69c85c070ca4..7869815c24037 100644
--- a/pandas/tests/frame/test_analytics.py
+++ b/pandas/tests/frame/test_analytics.py
@@ -1258,3 +1258,30 @@ def test_min_max_dt64_with_NaT(self):
         res = df.max()
         exp = pd.Series([pd.NaT], index=["foo"])
         tm.assert_series_equal(res, exp)
+
+    def test_min_max_dt64_api_consistency_with_NaT(self):
+        # Calling the following sum functions returned an error for dataframes but
+        # returned NaT for series. These tests check that the API is consistent in
+        # min/max calls on empty Series/DataFrames. See GH:33704 for more
+        # information
+        df = pd.DataFrame(dict(x=pd.to_datetime([])))
+        expected_dt_series = pd.Series(pd.to_datetime([]))
+        # check axis 0
+        assert (df.min(axis=0).x is pd.NaT) == (expected_dt_series.min() is pd.NaT)
+        assert (df.max(axis=0).x is pd.NaT) == (expected_dt_series.max() is pd.NaT)
+
+        # check axis 1
+        tm.assert_series_equal(df.min(axis=1), expected_dt_series)
+        tm.assert_series_equal(df.max(axis=1), expected_dt_series)
+
+    def test_min_max_dt64_api_consistency_empty_df(self):
+        # check DataFrame/Series api consistency when calling min/max on an empty
+        # DataFrame/Series.
+        df = pd.DataFrame(dict(x=[]))
+        expected_float_series = pd.Series([], dtype=float)
+        # check axis 0
+        assert np.isnan(df.min(axis=0).x) == np.isnan(expected_float_series.min())
+        assert np.isnan(df.max(axis=0).x) == np.isnan(expected_float_series.max())
+        # check axis 1
+        tm.assert_series_equal(df.min(axis=1), expected_float_series)
+        tm.assert_series_equal(df.min(axis=1), expected_float_series)