Gareth Bale opened the scoring in the second half but Alexander Tettey and Simeon Jackson entered the fray as substitutes to grab the goals that put the hosts through
Gareth Bale had given the visitors the lead after the break with a 20-yard shot that found the bottom corner and beat the reach of the otherwise excellent Mark Bunn.
But Norwich showed great spirit to secure a place in the quarter-final as Tettey fired home the equaliser before Grant Holt, another of Hughton's introductions, forced a save that left Jackson with a simple finish.
Andre Villas-Boas' team had a chance to send the game to extra time when Kyle Walker was fouled by Marc Tierney in the box but Clint Dempsey's penalty was saved by Bunn.
Dempsey led the line for Tottenham while Bale also started, but Aaron Lennon and Jermain Defoe both took a place on the bench. Hugo Lloris, Michael Dawson and Kyle Naughton were brought in at the back, with Brad Friedel, Jan Vertonghen and William Gallas rested.
Norwich boss Hughton, meanwhile, handed a competitive debut to Steven Whittaker and a second start to fellow new signing Jacob Butterfield.
Whittaker faced the daunting task of stopping Bale on his debut but after 15 quiet minutes, the Welsh winger was switched to the right flank and became Tierney's responsibility. It was at that point that Spurs first began to ask questions of the hosts' back four, with Dempsey prodding a shot narrowly over the bar after Bale had supplied a through pass.
Bale was lively throughout but the first period was, on the whole, one of half-chances as both sides struggled to work the space needed in midfield to gain any attacking rhythm. Lloris and Bunn, both aiming to unseat established goalkeepers at their respective clubs, looked assured when the likes of Bale and Morison did threaten.
The game was more open after the break. Andrew Surman was twice denied a shot on goal by good tackling inside the Tottenham penalty area and Iago Falque freed Bale with a measured through ball but Bunn comfortably saved.
Falque was growing in influence and forced Bunn into his best save yet when he cut inside and curled a shot towards the far corner. Lloris continued to match his counterpart by saving at Morison's feet and punching well from the subsequent corner.
A moment of individual magic always looked the most likely route to goal and Bale was always the most likely man to provide it. His opening strike was simple enough; lacking passing options, the 23-year-old turned and glided past David Fox before drilling a low, right-footed shot beyond Bunn from 25 yards.
Robert Snodgrass teed up Jonny Howson from similar range but Lloris had a clean sheet in sight and parried strongly. Unfortunately, there was nothing he could do as Norwich roared into life in an astonishing final 15 minutes.
Tettey had been introduced moments before Lloris' best save and the Norwegian, who is yet to lose a game with the Canaries, inspired the comeback. Substitute Vertonghen could only head a corner as far clear as Tettey and his left-footed volley found the bottom corner with the help of a deflection off the Belgian's leg.
Norwich took the initiative and added another to dispatch another of north London's clubs at Carrow Road. Snodgrass' free-kick from the left was headed towards goal by another replacement, Holt, and while Lloris denied the captain's effort he could do nothing to stop Jackson slamming home the rebound.
Spurs were handed a lifeline when Tierney fouled Kyle Walker as he charged into the box but Dempsey's penalty was weakly-struck and Bunn capped off a fine display with a crucial save.