diff --git a/Lib/test/regrtest.py b/Lib/test/regrtest.py
index e1d3f7f7e34..92a3d11cea2 100755
--- a/Lib/test/regrtest.py
+++ b/Lib/test/regrtest.py
@@ -923,6 +923,10 @@ def runtest_inner(test, verbose, quiet,
 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