diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 01cdca7aecf..fb9d9dfa910 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -94,6 +94,10 @@ Bug fixes
 - Masking data arrays with :py:meth:`xarray.DataArray.where` now returns an
   array with the name of the original masked array (:issue:`2748` and :issue:`2457`).
   By `Yohai Bar-Sinai <https://github.com/yohai>`_.
+- Fixed error when trying to reduce a DataArray using a function which does not
+  require an axis argument. (:issue:`2768`)
+  By `Tom Nicholas <http://github.com/TomNicholas>`_.
+
 - Per `CF conventions
   <http://cfconventions.org/cf-conventions/cf-conventions.html#calendar>`_,
   specifying ``'standard'`` as the calendar type in
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index 85eab294619..b675317d83d 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -1361,8 +1361,11 @@ def reduce(self, func, dim=None, axis=None,
 
         if dim is not None:
             axis = self.get_axis_num(dim)
-        data = func(self.data if allow_lazy else self.values,
-                    axis=axis, **kwargs)
+        input_data = self.data if allow_lazy else self.values
+        if axis is not None:
+            data = func(input_data, axis=axis, **kwargs)
+        else:
+            data = func(input_data, **kwargs)
 
         if getattr(data, 'shape', ()) == self.shape:
             dims = self.dims
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index e4ffdad4260..c26968b1db0 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -3633,11 +3633,28 @@ def mean_only_one_axis(x, axis):
         actual = ds.reduce(mean_only_one_axis, 'y')
         assert_identical(expected, actual)
 
-        with raises_regex(TypeError, 'non-integer axis'):
+        with raises_regex(TypeError, "missing 1 required positional argument: "
+                                     "'axis'"):
             ds.reduce(mean_only_one_axis)
 
         with raises_regex(TypeError, 'non-integer axis'):
-            ds.reduce(mean_only_one_axis, ['x', 'y'])
+            ds.reduce(mean_only_one_axis, axis=['x', 'y'])
+
+    def test_reduce_no_axis(self):
+
+        def total_sum(x):
+            return np.sum(x.flatten())
+
+        ds = Dataset({'a': (['x', 'y'], [[0, 1, 2, 3, 4]])})
+        expected = Dataset({'a': ((), 10)})
+        actual = ds.reduce(total_sum)
+        assert_identical(expected, actual)
+
+        with raises_regex(TypeError, "unexpected keyword argument 'axis'"):
+            ds.reduce(total_sum, axis=0)
+
+        with raises_regex(TypeError, "unexpected keyword argument 'axis'"):
+            ds.reduce(total_sum, dim='x')
 
     def test_quantile(self):