diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 39ba72063ef..f6fdcef9306 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -56,6 +56,10 @@ Bug fixes
   `Spencer Clark <https://github.com/spencerkclark>`_.
 - Line plots with the `x` argument set to a non-dimensional coord now plot the correct data for 1D DataArrays.
   (:issue:`27251). By `Tom Nicholas <http://github.com/TomNicholas>`_.
+- Subtracting a scalar ``cftime.datetime`` object from a
+  :py:class:`CFTimeIndex` now results in a :py:class:`pandas.TimedeltaIndex`
+  instead of raising a ``TypeError`` (:issue:`2671`).  By `Spencer Clark
+  <https://github.com/spencerkclark>`_.
 
 .. _whats-new.0.11.3:
 
diff --git a/xarray/coding/cftimeindex.py b/xarray/coding/cftimeindex.py
index 1861d49a1d4..f1a05d31a0c 100644
--- a/xarray/coding/cftimeindex.py
+++ b/xarray/coding/cftimeindex.py
@@ -408,13 +408,17 @@ def __radd__(self, other):
         return CFTimeIndex(other + np.array(self))
 
     def __sub__(self, other):
-        if isinstance(other, CFTimeIndex):
+        import cftime
+        if isinstance(other, (CFTimeIndex, cftime.datetime)):
             return pd.TimedeltaIndex(np.array(self) - np.array(other))
         elif isinstance(other, pd.TimedeltaIndex):
             return CFTimeIndex(np.array(self) - other.to_pytimedelta())
         else:
             return CFTimeIndex(np.array(self) - other)
 
+    def __rsub__(self, other):
+        return pd.TimedeltaIndex(other - np.array(self))
+
     def _add_delta(self, deltas):
         # To support TimedeltaIndex + CFTimeIndex with older versions of
         # pandas.  No longer used as of pandas 0.23.
diff --git a/xarray/tests/test_cftimeindex.py b/xarray/tests/test_cftimeindex.py
index 0d6ba6b47c9..97be993d842 100644
--- a/xarray/tests/test_cftimeindex.py
+++ b/xarray/tests/test_cftimeindex.py
@@ -700,6 +700,26 @@ def test_cftimeindex_sub_cftimeindex(calendar):
     assert isinstance(result, pd.TimedeltaIndex)
 
 
+@pytest.mark.skipif(not has_cftime, reason='cftime not installed')
+@pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
+def test_cftimeindex_sub_cftime_datetime(calendar):
+    a = xr.cftime_range('2000', periods=5, calendar=calendar)
+    result = a - a[0]
+    expected = pd.TimedeltaIndex([timedelta(days=i) for i in range(5)])
+    assert result.equals(expected)
+    assert isinstance(result, pd.TimedeltaIndex)
+
+
+@pytest.mark.skipif(not has_cftime, reason='cftime not installed')
+@pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
+def test_cftime_datetime_sub_cftimeindex(calendar):
+    a = xr.cftime_range('2000', periods=5, calendar=calendar)
+    result = a[0] - a
+    expected = pd.TimedeltaIndex([timedelta(days=-i) for i in range(5)])
+    assert result.equals(expected)
+    assert isinstance(result, pd.TimedeltaIndex)
+
+
 @pytest.mark.skipif(not has_cftime, reason='cftime not installed')
 @pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
 def test_cftimeindex_sub_timedeltaindex(calendar):