Sami Hyypia put out a first-choice XI at the BayArena, but the 2. Bundesliga promotion chasers held firm, forcing extra time - in which Ruben Yttergard Jenssen struck to seal the shock of the round.
The visitors were on the back foot for much of the contest, but they were resolute as Leverkusen failed to conjure up anything impressive enough to break their lower-ranked opponents down.
As extra time loomed, Kaiserslautern found their threat on the counter and were awarded a penalty as Leverkusen debutant Andres Guardado brought Karim Matmour down, but Mohammadou Idrissou's weak effort was easily saved by Bernd Leno.
Kroos was again heavily involved in the visitors' third, dinking a neat ball over Hamburg's defence that Arjen Robben raced on to before finishing.
Mandzukic's second was again provided by Gotze - whose cross-cum-shot found the Croatian in space.
The striker's hat-trick - and the scoring in the tie - was wrapped up with 15 minutes to go as sloppy home defending gave him the chance to smash home.
Ricardo Rodriguez scored twice - both from the penalty spot - to sandwich Roberto Firmino's first-half equaliser.
Slack marking allowed Bas Dost to prod home on the hour, with Firmino's consolation in stoppage time counting for little.
