--- a/testing/web-platform/update/upstream.py
+++ b/testing/web-platform/update/upstream.py
@@ -21,21 +21,22 @@ def rewrite_patch(patch, strip_dir):
     :param patch: the Patch to convert
     :param strip_dir: the path prefix to remove
     """
 
     if not strip_dir.startswith("/"):
         strip_dir = "/%s"% strip_dir
 
     new_diff = []
-    line_starts = ["diff ", "+++ ", "--- ", "rename from ", "rename to "]
+    line_starts = [("diff ", True), ("+++ ", True), ("--- ", True), ("rename from ", False), ("rename to ", False)]
     for line in patch.diff.split("\n"):
-        for start in line_starts:
+        for start, leading_slash in line_starts:
+            strip = strip_dir if leading_slash else strip_dir[1:]
             if line.startswith(start):
-                new_diff.append(line.replace(strip_dir, "").encode("utf8"))
+                new_diff.append(line.replace(strip, "").encode("utf8"))
                 break
         else:
             new_diff.append(line)
 
     new_diff = "\n".join(new_diff)
 
     assert new_diff != patch