Gareth Bale’s second goal in as many games looked to have secured Spurs’ progress at Carrow Road, but a late own goal by Jan Vertonghen and Simeon Jackson strike put the Canaries in front with three minutes remaining.
The visitors were awarded a penalty at the death when Marc Tierney felled Kyle Walker, but goalkeeper Mark Bunn capped off a superb performance by saving Clint Dempsey’s penalty.
Such late fireworks looked unlikely after a forgettable first half, where both teams struggled for fluency after making a host of changes.
Spurs were the most threatening, Bale central and Dempsey seeing a couple of shots saved by Bunn.
Norwich’s best chance of the half fell to Steve Morison on 12 minutes when he met Robert Snodgrass’ corner, but Hugo Lloris saved his header.
Bunn’ heroics ensured Norwich went in level at the break and he continued that form with fantastic stops from Bale and Falque.
However, the 27-year-old was given no chance on 65 minutes when Bale broke the deadlock with a low drive.
Norwich threw on Grant Holt as they chased an equaliser, but it came from another source when Tettey’s shot deflected in off Vertonghen.