diff --git a/include/mxnet/c_api.h b/include/mxnet/c_api.h
index 4dd858a51c4b..5837069a8c32 100644
--- a/include/mxnet/c_api.h
+++ b/include/mxnet/c_api.h
@@ -2317,6 +2317,11 @@ MXNET_DLL int MXNDArrayGetSharedMemHandle(NDArrayHandle handle, int* shared_pid,
 MXNET_DLL int MXNDArrayCreateFromSharedMem(int shared_pid, int shared_id, const mx_uint *shape,
                                            mx_uint ndim, int dtype, NDArrayHandle *out);
 
+/*!
+ * \brief Query cuDNN for minimum permissible epsilon for BatchNorm. If not installed, return NaN.
+ * \param result the minimum epsilon provided by cuDNN
+ */
+MXNET_DLL int MXGetCudnnBnEpsilon(float* result);
 
 #ifdef __cplusplus
 }
diff --git a/python/mxnet/base.py b/python/mxnet/base.py
index 0fb73b3c7dda..0b0c4a8e8e35 100644
--- a/python/mxnet/base.py
+++ b/python/mxnet/base.py
@@ -680,3 +680,16 @@ def write_all_str(module_file, module_all_list):
     module_op_file.close()
     write_all_str(module_internal_file, module_internal_all)
     module_internal_file.close()
+
+def get_cudnn_epsilon():
+    """Check value of CUDNN_BN_MIN_EPSILON. If the value
+       of epsilon in a model is less than this, cuDNN BatchNorm
+       should be replaced with MxNet BatchNorm since cuDNN requires
+       this minimum to be met. This can be helpful e.g. in ONNX import,
+       so that cuDNN BN is only disabled if the value in the checkpoint
+       is less than CUDNN_BN_MIN_EPSILON, and enabled otherwise.
+       If cuDNN is not installed, this call with return NAN instead of
+       an exception."""
+    result = ctypes.c_float(0.0)
+    _LIB.MXGetCudnnBnEpsilon(ctypes.byref(result))
+    return result.value
diff --git a/python/mxnet/contrib/onnx/onnx2mx/_op_translations.py b/python/mxnet/contrib/onnx/onnx2mx/_op_translations.py
index 2b98aa08febf..27553cff282e 100644
--- a/python/mxnet/contrib/onnx/onnx2mx/_op_translations.py
+++ b/python/mxnet/contrib/onnx/onnx2mx/_op_translations.py
@@ -18,9 +18,11 @@
 # coding: utf-8
 """ Module for translating ONNX operators into Mxnet operatoes"""
 # pylint: disable=unused-argument,protected-access
+import math
 import numpy as np
 from . import _translation_utils as translation_utils
 from .... import symbol
+from ....base import get_cudnn_epsilon
 
 # Method definitions for the callable objects mapped in the import_helper module
 
@@ -209,7 +211,9 @@ def batch_norm(attrs, inputs, proto_obj):
                                                                'is_test': 'fix_gamma'})
     new_attrs = translation_utils._remove_attributes(new_attrs,
                                                      ['spatial', 'consumed_inputs'])
-    new_attrs = translation_utils._add_extra_attributes(new_attrs, {'cudnn_off': 1})
+    cudnn_eps = get_cudnn_epsilon()
+    cudnn_off = 0 if not math.isnan(cudnn_eps) and attrs.get('epsilon', 1e-5) >= cudnn_eps else 1
+    new_attrs = translation_utils._add_extra_attributes(new_attrs, {'cudnn_off': cudnn_off})
 
     # in test mode "fix_gamma" should be unset.
     new_attrs['fix_gamma'] = not attrs.get('is_test', 1)
diff --git a/src/c_api/c_api.cc b/src/c_api/c_api.cc
index 467118b9921e..3936cc302a0c 100644
--- a/src/c_api/c_api.cc
+++ b/src/c_api/c_api.cc
@@ -36,6 +36,7 @@
 #include <mxnet/kvstore.h>
 #include <mxnet/rtc.h>
 #include <mxnet/storage.h>
+#include <cmath>
 #include <vector>
 #include <sstream>
 #include <string>
@@ -47,6 +48,10 @@
 #include "../operator/custom/custom-inl.h"
 #include "../operator/tensor/matrix_op-inl.h"
 
+#if MXNET_USE_CUDA && MXNET_USE_CUDNN
+#include <cudnn.h>
+#endif
+
 using namespace mxnet;
 
 // Internal function to get the information
@@ -1312,3 +1317,13 @@ int MXNDArrayCreateFromSharedMem(int shared_pid, int shared_id, const mx_uint *s
   *out = new NDArray(shared_pid, shared_id, TShape(shape, shape + ndim), dtype);
   API_END();
 }
+
+int MXGetCudnnBnEpsilon(float* result) {
+  API_BEGIN();
+#if MXNET_USE_CUDA && MXNET_USE_CUDNN && defined(CUDNN_BN_MIN_EPSILON)
+    *result = static_cast<float>(CUDNN_BN_MIN_EPSILON);
+#else
+    *result = NAN;
+#endif  // MXNET_USE_CUDA && MXNET_USE_CUDNN && defined(CUDNN_BN_MIN_EPSILON)
+  API_END();
+}
diff --git a/tests/python/gpu/test_cudnn_eps.py b/tests/python/gpu/test_cudnn_eps.py
new file mode 100644
index 000000000000..5b3ea2008059
--- /dev/null
+++ b/tests/python/gpu/test_cudnn_eps.py
@@ -0,0 +1,31 @@
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements.  See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership.  The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License.  You may obtain a copy of the License at
+#
+#   http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied.  See the License for the
+# specific language governing permissions and limitations
+# under the License.
+
+import math
+import mxnet as mx
+import numpy as np
+import unittest
+
+
+def test_get_cudnn_epsilon():
+    eps = mx.base.get_cudnn_epsilon()
+    assert math.isnan(eps) or np.isclose(1e-5, mx.base.get_cudnn_epsilon(), atol=1e-10), \
+        "cudnn eps is non NaN and it's not close to 1e-5"
+    print ("Passed")
+
+if __name__ == '__main__':
+    test_get_cudnn_epsilon()