diff --git a/cloudpickle/cloudpickle.py b/cloudpickle/cloudpickle.py
index e8f4223b2..fb40342e8 100644
--- a/cloudpickle/cloudpickle.py
+++ b/cloudpickle/cloudpickle.py
@@ -238,7 +238,7 @@ def save_function(self, obj, name=None):
         # a builtin_function_or_method which comes in as an attribute of some
         # object (e.g., object.__new__, itertools.chain.from_iterable) will end
         # up with modname "__main__" and so end up here. But these functions
-        # have no __code__ attribute in CPython, so the handling for 
+        # have no __code__ attribute in CPython, so the handling for
         # user-defined functions below will fail.
         # So we pickle them here using save_reduce; have to do it differently
         # for different python versions.
@@ -282,6 +282,27 @@ def save_function(self, obj, name=None):
             self.memoize(obj)
     dispatch[types.FunctionType] = save_function
 
+    def _save_subimports(self, code, top_level_dependencies):
+        """
+        Ensure de-pickler imports any package child-modules that
+        are needed by the function
+        """
+        # check if any known dependency is an imported package
+        for x in top_level_dependencies:
+            if isinstance(x, types.ModuleType) and x.__package__:
+                # check if the package has any currently loaded sub-imports
+                prefix = x.__name__ + '.'
+                for name, module in sys.modules.items():
+                    if name.startswith(prefix):
+                        # check whether the function can address the sub-module
+                        tokens = set(name[len(prefix):].split('.'))
+                        if not tokens - set(code.co_names):
+                            # ensure unpickler executes this import
+                            self.save(module)
+                            # then discards the reference to it
+                            self.write(pickle.POP)
+
+
     def save_function_tuple(self, func):
         """  Pickles an actual func object.
 
@@ -307,6 +328,8 @@ def save_function_tuple(self, func):
         save(_fill_function)  # skeleton function updater
         write(pickle.MARK)    # beginning of tuple that _fill_function expects
 
+        self._save_subimports(code, set(f_globals.values()) | set(closure))
+
         # create a skeleton function object and memoize it
         save(_make_skel_func)
         save((code, closure, base_globals))
diff --git a/tests/cloudpickle_test.py b/tests/cloudpickle_test.py
index a1dec1151..c540d5d4c 100644
--- a/tests/cloudpickle_test.py
+++ b/tests/cloudpickle_test.py
@@ -9,6 +9,8 @@
 import itertools
 import platform
 import textwrap
+import base64
+import subprocess
 
 try:
     # try importing numpy and scipy. These are not hard dependencies and
@@ -360,6 +362,92 @@ def f():
         self.assertTrue(f2 is f3)
         self.assertEqual(f2(), res)
 
+    def test_submodule(self):
+        # Function that refers (by attribute) to a sub-module of a package.
+
+        # Choose any module NOT imported by __init__ of its parent package
+        # examples in standard library include:
+        # - http.cookies, unittest.mock, curses.textpad, xml.etree.ElementTree
+
+        global xml # imitate performing this import at top of file
+        import xml.etree.ElementTree
+        def example():
+            x = xml.etree.ElementTree.Comment # potential AttributeError
+
+        s = cloudpickle.dumps(example)
+
+        # refresh the environment, i.e., unimport the dependency
+        del xml
+        for item in list(sys.modules):
+            if item.split('.')[0] == 'xml':
+                del sys.modules[item]
+
+        # deserialise
+        f = pickle.loads(s)
+        f() # perform test for error
+
+    def test_submodule_closure(self):
+        # Same as test_submodule except the package is not a global
+        def scope():
+            import xml.etree.ElementTree
+            def example():
+                x = xml.etree.ElementTree.Comment # potential AttributeError
+            return example
+        example = scope()
+
+        s = cloudpickle.dumps(example)
+
+        # refresh the environment (unimport dependency)
+        for item in list(sys.modules):
+            if item.split('.')[0] == 'xml':
+                del sys.modules[item]
+
+        f = cloudpickle.loads(s)
+        f() # test
+
+    def test_multiprocess(self):
+        # running a function pickled by another process (a la dask.distributed)
+        def scope():
+            import curses.textpad
+            def example():
+                x = xml.etree.ElementTree.Comment
+                x = curses.textpad.Textbox
+            return example
+        global xml
+        import xml.etree.ElementTree
+        example = scope()
+
+        s = cloudpickle.dumps(example)
+
+        # choose "subprocess" rather than "multiprocessing" because the latter
+        # library uses fork to preserve the parent environment.
+        command = ("import pickle, base64; "
+                   "pickle.loads(base64.b32decode('" +
+                   base64.b32encode(s).decode('ascii') +
+                   "'))()")
+        assert not subprocess.call([sys.executable, '-c', command])
+
+    def test_import(self):
+        # like test_multiprocess except subpackage modules referenced directly
+        # (unlike test_submodule)
+        global etree
+        def scope():
+            import curses.textpad as foobar
+            def example():
+                x = etree.Comment
+                x = foobar.Textbox
+            return example
+        example = scope()
+        import xml.etree.ElementTree as etree
+
+        s = cloudpickle.dumps(example)
+
+        command = ("import pickle, base64; "
+                   "pickle.loads(base64.b32decode('" +
+                   base64.b32encode(s).decode('ascii') +
+                   "'))()")
+        assert not subprocess.call([sys.executable, '-c', command])
+
 
 if __name__ == '__main__':
     unittest.main()