diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 93166759d8dbd..6e3c579ed7b17 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -299,7 +299,26 @@ class TimelikeOps:
     def _round(self, freq, mode, ambiguous, nonexistent):
         # round the local times
         values = _ensure_datetimelike_to_i8(self)
-        result = round_nsint64(values, mode, freq)
+        try:
+            result = round_nsint64(values, mode, freq)
+        except ValueError as e:
+            if "non-fixed" in str(e):
+                offset = frequencies.to_offset(freq)
+                if mode == RoundTo.PLUS_INFTY:
+                    result = self.to_period(offset) \
+                        .to_timestamp(how='end').asi8
+                elif mode == RoundTo.MINUS_INFTY:
+                    result = self.to_period(offset) \
+                        .to_timestamp(how='start').asi8
+                elif mode == RoundTo.NEAREST_HALF_EVEN:
+                    msg = ("round only supports fixed offsets "
+                           "(i.e. 'Day' is ok, 'MonthEnd' is not). "
+                           "You can use dti.snap or floor/ceil if "
+                           "applicable.")
+                    raise ValueError(msg)
+                else:
+                    raise e
+
         result = self._maybe_mask_results(result, fill_value=NaT)
 
         dtype = self.dtype
diff --git a/pandas/tests/arrays/test_datetimes.py b/pandas/tests/arrays/test_datetimes.py
index c7c0e1180ce46..1c1990a037ee7 100644
--- a/pandas/tests/arrays/test_datetimes.py
+++ b/pandas/tests/arrays/test_datetimes.py
@@ -11,9 +11,10 @@
 import pandas as pd
 from pandas.core.arrays import DatetimeArray
 from pandas.core.arrays.datetimes import sequence_to_dt64ns
+from pandas.tseries.offsets import Nano
+from pandas import Timestamp
 import pandas.util.testing as tm
 
-
 class TestDatetimeArrayConstructor:
 
     def test_only_1dim_accepted(self):
@@ -301,3 +302,98 @@ def test_min_max_empty(self, skipna, tz):
 
         result = arr.max(skipna=skipna)
         assert result is pd.NaT
+
+    @pytest.mark.parametrize(
+        'op, freq, dates, expected_dates',
+        [
+            (
+                'floor',
+                'M',
+                ("2001-02-01",
+                 Timestamp("2001-02-14 12:00") - Nano(),
+                 "2001-02-14 12:00",
+                 "2001-02-15",
+                 Timestamp("2001-03-01") - Nano(),
+                 "2001-03-01"
+                 ),
+                [
+                    Timestamp("2001-02-01") - Nano(),
+                    Timestamp("2001-02-01") - Nano(),
+                    Timestamp("2001-02-01") - Nano(),
+                    Timestamp("2001-02-01") - Nano(),
+                    Timestamp("2001-03-01") - Nano(),
+                    Timestamp("2001-03-01") - Nano(),
+                ]
+            ),
+            (
+                'ceil',
+                'M',
+                ("2001-02-01",
+                 Timestamp(
+                     "2001-02-14 12:00") - Nano(),
+                 "2001-02-14 12:00",
+                 "2001-02-15",
+                 Timestamp("2001-03-01") - Nano(),
+                 "2001-03-01"
+                 ),
+                [
+                    Timestamp("2001-03-01") - Nano(),
+                    Timestamp("2001-03-01") - Nano(),
+                    Timestamp("2001-03-01") - Nano(),
+                    Timestamp("2001-03-01") - Nano(),
+                    Timestamp("2001-03-01") - Nano(),
+                    Timestamp("2001-04-01") - Nano(),
+                ]
+            ),
+            (
+                'floor',
+                'MS',
+                ("2001-02-01",
+                 Timestamp("2001-02-14 12:00") - Nano(),
+                 "2001-02-14 12:00",
+                 "2001-02-15",
+                 Timestamp("2001-03-01") - Nano(),
+                 "2001-03-01"
+                 ),
+                [
+                    "2001-02-01",
+                    "2001-02-01",
+                    "2001-02-01",
+                    "2001-02-01",
+                    "2001-02-01",
+                    "2001-03-01"
+                ]
+            ),
+            (
+                'ceil',
+                'MS',
+                ("2001-02-01",
+                 Timestamp("2001-02-14 12:00") - Nano(),
+                 "2001-02-14 12:00",
+                 "2001-02-15",
+                 Timestamp("2001-03-01") - Nano(),
+                 "2001-03-01"
+                 ),
+                [
+                    "2001-02-01",
+                    "2001-03-01",
+                    "2001-03-01",
+                    "2001-03-01",
+                    "2001-03-01",
+                    "2001-03-01",
+                ]
+            )
+        ]
+    )
+    def test_ceil_floor(self, op, freq, dates, expected_dates):
+        dta = DatetimeArray._from_sequence(dates)
+        dta[1] -= pd.offsets.Nano()
+        dta[-2] -= pd.offsets.Nano()
+        dti = pd.DatetimeIndex(dta)
+        result = getattr(dti, op)(freq)
+        expected = pd.DatetimeIndex(expected_dates)
+        tm.assert_index_equal(result, expected)
+
+        # Idempotent
+        result = getattr(result, op)(freq)
+        tm.assert_index_equal(result, expected)