diff --git a/xarray/core/indexing.py b/xarray/core/indexing.py
index ab049a0a4b4..87a53d8c879 100644
--- a/xarray/core/indexing.py
+++ b/xarray/core/indexing.py
@@ -205,27 +205,35 @@ def convert_label_indexer(index, label, index_name="", method=None, tolerance=No
 
 def get_dim_indexers(data_obj, indexers):
     """Given a xarray data object and label based indexers, return a mapping
-    of label indexers with only dimension names as keys.
+    of label indexers with only dimension names (or 1D non-dimensional coord
+    names) as keys.
 
     It groups multiple level indexers given on a multi-index dimension
     into a single, dictionary indexer for that dimension (Raise a ValueError
     if it is not possible).
     """
+    non_dim_1d_coords = [coord for coord in data_obj.coords
+                         if len(data_obj[coord].dims) == 1]
     invalid = [
-        k
-        for k in indexers
-        if k not in data_obj.dims and k not in data_obj._level_coords
+        k for k in indexers
+           if k not in data_obj.dims and k not in data_obj._level_coords
+             and k not in non_dim_1d_coords
     ]
-    if invalid:
-        raise ValueError(f"dimensions or multi-index levels {invalid!r} do not exist")
+    if invalid:  # TODO This was never covered by testing
+         raise ValueError(f"dimensions, 1D coordinates, or  multi-index levels"
+                          f" {invalid!r} do not exist")
 
     level_indexers = defaultdict(dict)
     dim_indexers = {}
     for key, label in indexers.items():
         (dim,) = data_obj[key].dims
         if key != dim:
-            # assume here multi-index level indexer
-            level_indexers[dim][key] = label
+            # If key is 1D non-dimension coordinate let it pass through
+            if key in data_obj.coords and data_obj.coords[key].dims == (dim,):
+                dim_indexers[key] = label
+            else:
+                # assume here multi-index level indexer
+                level_indexers[dim][key] = label
         else:
             dim_indexers[key] = label
 
@@ -253,9 +261,31 @@ def remap_label_indexers(data_obj, indexers, method=None, tolerance=None):
 
     dim_indexers = get_dim_indexers(data_obj, indexers)
     for dim, label in dim_indexers.items():
-        try:
+        if dim in data_obj.indexes:
             index = data_obj.indexes[dim]
-        except KeyError:
+
+            coords_dtype = data_obj.coords[dim].dtype
+            label = maybe_cast_to_coords_dtype(label, coords_dtype)
+            idxr, new_idx = convert_label_indexer(index, label, dim,
+                                                  method, tolerance)
+            pos_indexers[dim] = idxr
+            if new_idx is not None:
+                new_indexes[dim] = new_idx
+
+        elif dim in data_obj.coords and len(data_obj.coords[dim] == 1):
+            # 1D non-dimension coord
+            index = data_obj.coords[dim].to_index()
+            (dim,) = data_obj[dim].dims
+
+            coords_dtype = data_obj.coords[dim].dtype
+            label = maybe_cast_to_coords_dtype(label, coords_dtype)
+            idxr, new_idx = convert_label_indexer(index, label, dim,
+                                                  method, tolerance)
+            pos_indexers[dim] = idxr
+            if new_idx is not None:
+                new_indexes[dim] = new_idx
+
+        else:
             # no index for this dimension: reuse the provided labels
             if method is not None or tolerance is not None:
                 raise ValueError(
@@ -264,13 +294,6 @@ def remap_label_indexers(data_obj, indexers, method=None, tolerance=None):
                     "an associated coordinate."
                 )
             pos_indexers[dim] = label
-        else:
-            coords_dtype = data_obj.coords[dim].dtype
-            label = maybe_cast_to_coords_dtype(label, coords_dtype)
-            idxr, new_idx = convert_label_indexer(index, label, dim, method, tolerance)
-            pos_indexers[dim] = idxr
-            if new_idx is not None:
-                new_indexes[dim] = new_idx
 
     return pos_indexers, new_indexes
 
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index 237c315583c..833c41b9675 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -56,7 +56,7 @@
     pass
 
 
-def create_test_data(seed=None):
+def create_test_data(seed=None, dim_coords=True):
     rs = np.random.RandomState(seed)
     _vars = {
         "var1": ["dim1", "dim2"],
@@ -66,9 +66,14 @@ def create_test_data(seed=None):
     _dims = {"dim1": 8, "dim2": 9, "dim3": 10}
 
     obj = Dataset()
-    obj["time"] = ("time", pd.date_range("2000-01-01", periods=20))
-    obj["dim2"] = ("dim2", 0.5 * np.arange(_dims["dim2"]))
-    obj["dim3"] = ("dim3", list("abcdefghij"))
+    if dim_coords:
+        obj["time"] = ("time", pd.date_range("2000-01-01", periods=20))
+        obj["dim2"] = ("dim2", 0.5 * np.arange(_dims["dim2"]))
+        obj["dim3"] = ("dim3", list("abcdefghij"))
+    else:
+        obj.coords["time_coord"] = ("time", pd.date_range("2000-01-01", periods=20))
+        obj.coords["coord2"] = ("dim2", 0.5 * np.arange(_dims["dim2"]))
+        obj.coords["coord3"] = ("dim3", list("abcdefghij"))
     for v, dims in sorted(_vars.items()):
         data = rs.normal(size=tuple(_dims[d] for d in dims))
         obj[v] = (dims, data, {"foo": "variable"})
@@ -1368,6 +1373,16 @@ def test_sel_dataarray(self):
         assert_equal(actual.drop_vars("new_dim"), expected)
         assert np.allclose(actual["new_dim"].values, ind["new_dim"].values)
 
+    def test_sel_non_index_coord(self):
+        data = create_test_data(dim_coords=False)
+        int_slicers = {"dim2": slice(2),
+                       "dim3": slice(3)}
+        loc_slicers = {
+            "coord2": slice(0, 0.5),
+            "coord3": slice("a", "c"),
+        }
+        assert_equal(data.isel(**int_slicers), data.sel(**loc_slicers))
+
     def test_sel_dataarray_mindex(self):
         midx = pd.MultiIndex.from_product([list("abc"), [0, 1]], names=("one", "two"))
         mds = xr.Dataset(