diff --git a/numpy/core/einsumfunc.py b/numpy/core/einsumfunc.py
index 9ad0c7e1788d..da78748a3501 100644
--- a/numpy/core/einsumfunc.py
+++ b/numpy/core/einsumfunc.py
@@ -706,10 +706,13 @@ def einsum_path(*operands, **kwargs):
         for cnum, char in enumerate(term):
             dim = sh[cnum]
             if char in dimension_dict.keys():
-                if dimension_dict[char] != dim:
-                    raise ValueError("Size of label '%s' for operand %d does "
-                                     "not match previous terms."
-                                     % (char, tnum))
+                # For broadcasting cases we always want the largest dim size
+                if dimension_dict[char] == 1:
+                    dimension_dict[char] = dim
+                elif dim not in (1, dimension_dict[char]):
+                    raise ValueError("Size of label '%s' for operand %d (%d) "
+                                     "does not match previous terms (%d)."
+                                     % (char, tnum, dimension_dict[char], dim))
             else:
                 dimension_dict[char] = dim
 
@@ -1101,13 +1104,22 @@ def einsum(*operands, **kwargs):
         if specified_out and ((num + 1) == len(contraction_list)):
             handle_out = True
 
-        # Call tensordot
+        # Handle broadcasting vs BLAS cases
         if blas:
-
             # Checks have already been handled
             input_str, results_index = einsum_str.split('->')
             input_left, input_right = input_str.split(',')
-
+            if 1 in tmp_operands[0] or 1 in tmp_operands[1]:
+                left_dims = {dim: size for dim, size in
+                             zip(input_left, tmp_operands[0].shape)}
+                right_dims = {dim: size for dim, size in
+                              zip(input_right, tmp_operands[1].shape)}
+                # If dims do not match we are broadcasting, BLAS off
+                if any(left_dims[ind] != right_dims[ind] for ind in idx_rm):
+                    blas = False
+
+        # Call tensordot if still possible
+        if blas:
             tensor_result = input_left + input_right
             for s in idx_rm:
                 tensor_result = tensor_result.replace(s, "")
diff --git a/numpy/core/tests/test_einsum.py b/numpy/core/tests/test_einsum.py
index d07256ec291a..9bd85fdb9947 100644
--- a/numpy/core/tests/test_einsum.py
+++ b/numpy/core/tests/test_einsum.py
@@ -481,6 +481,25 @@ def check_einsum_sums(self, dtype, do_opt=False):
         r = np.arange(4).reshape(2, 2) + 7
         assert_equal(np.einsum('z,mz,zm->', p, q, r), 253)
 
+        # singleton dimensions broadcast (gh-10343)
+        p = np.ones((10,2))
+        q = np.ones((1,2))
+        assert_array_equal(np.einsum('ij,ij->j', p, q, optimize=True),
+                           np.einsum('ij,ij->j', p, q, optimize=False))
+        assert_array_equal(np.einsum('ij,ij->j', p, q, optimize=True),
+                           [10.] * 2)
+
+        p = np.ones((1, 5))
+        q = np.ones((5, 5))
+        for optimize in (True, False):
+            assert_array_equal(np.einsum("...ij,...jk->...ik", p, p,
+                                         optimize=optimize),
+                               np.einsum("...ij,...jk->...ik", p, q,
+                                         optimize=optimize))
+            assert_array_equal(np.einsum("...ij,...jk->...ik", p, q,
+                                         optimize=optimize),
+                               np.full((1, 5), 5))
+
     def test_einsum_sums_int8(self):
         self.check_einsum_sums('i1')