diff --git a/Lib/test/regrtest.py b/Lib/test/regrtest.py
index cac3c8c4a79..5b1f4e22c88 100755
--- a/Lib/test/regrtest.py
+++ b/Lib/test/regrtest.py
@@ -688,6 +688,10 @@ def runtest_inner(test, generate, verbose, quiet, test_times,
 def cleanup_test_droppings(testname, verbose):
     import shutil
     import stat
+    import gc
+
+    # First kill any dangling references to open files etc.
+    gc.collect()
 
     # Try to clean up junk commonly left behind.  While tests shouldn't leave
     # any files or directories behind, when a test fails that can be tedious