Gareth Bale, who had torn Norwich apart here last season, seemed to have put Spurs on course for victory when he drove home a 20-yard shot in the second-half, but two goals in the last six minutes gave the Canaries an unlikely victory.
First substitute Alex Tettey's shot flew in off the out-stretched leg of Jan Vertonghen and then Simeon Jackson pounced on an error by Hugo Lloris to make it 2-1 with three minutes left.
The drama did not end there, however, as Kyle Walker was upended in the Norwich box by Marc Tierney a minute later.
Clint Dempsey stepped up to take the spot-kick, but Bunn, making just his second start of the season, saved the American's shot to gives the Canaries' season another valuable shot in the arm.
For much of the game, home manager Chris Hughton never looked like beating the club with whom he played for 14 years.
Apart from an early opportunity from Steve Morison, who was denied by a crucial block from Steven Caulker, the Canaries were the second best team for most of the first half.
After a slow opening 10 minutes from both sides Andre Villas-Boas shuffled his pack by swapping Bale on to the right to give him a run at Tierney.
The Norwich left-back struggled to keep his composure, mis-placing a header that allowed Bale to charge at the Canaries defence but he shot straight at Mark Bunn.
The Welshman got the better of Tierney moments later and found Dempsey, but the American's clever chip sailed just over Bunn's goal.
Bale then rippled the Norwich side-netting from a Jake Livermore cross and shot inches wide soon after as the winger threatened to open the home side up.
The 23-year-old was then in the thick of the action for the wrong reasons as his late sliding tackle on Tierney earned him a booking.
Jonny Howson flashed a volley two yards wide just before referee Jon Moss brought the dull opening half to a halt.
The second period started with much more purpose, with Spurs looking more threatening thanks to the introduction of Tom Huddlestone.
Norwich also looked more potent, however, and only a crucial block from Caulker denied Jacob Butterfield his first goal for the Canaries.
Falque almost caused Morison to put through his own goal thanks to a wicked cross and the Spaniard then played in Bale only to see the former Southampton man shoot straight at Bunn.
The stopper had to be at his best to deny Falque's brilliant curling shot, but Norwich then flew up the other end through Morison, who was denied by some quick thinking from Lloris.
Norwich were playing with much more confidence, but their hopes of an upset took a hit in the 66th minute when Bale latched on to Tom Carroll's pass and drove a sweet right-footed drive past the diving Bunn.
Hughton's team responded well. With Tettey and Jackson drafted on from the bench, the hosts pushed for an immediate equaliser and they almost got it when Howson's wicked shot needed an acrobatic save from Lloris.
Norwich piled on the pressure and Hughton took one last throw of the dice when he brought on captain Grant Holt.
The pressure told as the home side equalised, albeit with a huge slice of luck.
Robert Snodgrass' corner fell in to the path of Tettey, whose shot was going wide until Vertonghen inexplicably stuck a leg out to divert the ball past Lloris.
Tottenham's heads dropped, but worse was to come. With three minutes left Lloris spilled Holt's header straight at the feet of Jackson, who lashed past the Frenchman from close range.
There was still time for one last twist, however. Tierney clumsily felled Walker one minute later and Moss pointed to the spot.
Dempsey stepped up to take the spot-kick, but Bunn saved well to his left, causing ecstatic scenes among the home support.
Michael Dawson had a chance to take the game to extra-time in the last of four minutes off added time but once again Bunn saved to knock the Londoners out of the competition.