We looked in control of the fourth round tie after Gareth Bale fired us into the lead in the 66th minute but Norwich scored in the 83rd and 86th minutes to turn the game completely on its head.
In a dramatic finale, Kyle Walker was upended in the box two minutes from time, but home goalkeeper Mark Bunn guessed the right way and saved Clint Dempsey's penalty to his left.
Andre Villas-Boas made seven changes from the starting line-up at Southampton on Sunday.
Hugo Lloris came in for Brad Friedel in goal with Michael Dawson, Kyle Naughton, Tom Carroll, Jake Livermore, Yago Falque and Gylfi Sigurdsson also lining-up. It was Carroll's first start of the campaign, having appeared off the bench in our third round win at Carlisle. Clint Dempsey led the line.
We controlled the first half in terms of possession, but there were few clear-cut chances at either end.
Dempsey headed wide from Naughton's cross in the eighth minute and Bale had the next sight of goal, latching onto Marc Tierney's poor header and striking straight at Bunn from 22 yards.
Bale switched flanks with Falque early and was central to the best chance of the half on 23 minutes. He found room on halfway and took off towards goal and at the edge of the box, rolled the ball into Dempsey who flicked his chance over Bunn, but also inches over the crossbar.
Morison headed straight at Lloris from a tight angle on 26 minutes and play was soon at the other end as Livermore worked room on the right and drove to the near post, where Bale flicked into the side-netting.
Bale fired just wide from distance before Dempsey had the final effort of the half, a low curler from 20 yards that Bunn took comfortably down to his right.
We were forced into a change at half-time as Livermore, injured in a first half challenge, couldn't continue and was replaced by Tom Huddlestone.
Steven Caulker made a fine block from Andrew Surman but we were soon in control again.
Bale tested Bunn at his near post from Falque's fine pass and the goalkeeper did even better minutes later to turn Falque's curler from 22 yards around the post.
Lloris made his first serious save in the 64th minute, the French international out like a flash to block from Morison.
Two minutes later and we were in the lead. Carroll returned a throw to Naughton, he rolled into Bale's feet on the left edge of the box and Bale turned inside, away from Fox and drilled low into the bottom corner with his right foot from 20 yards.
Lloris tipped away Jonny Howson's strike but we still looked in control when Chris Hughton brought on Grant Holt as a last throw of the dice.
Within a minute of Holt's arrival, the Canaries found a fortuitous equaliser.
Vertonghen, on himself as substitute for Carroll with 11 minutes left, did well to head clear a Snodgrass corner under pressure from Holt, the ball fell to Alex Tettey and his volley was flying a yard wide when it clipped off Vertonghen and home.
Three minutes later and an unlikely turn around was complete as Michael Turner met Snodgrass's free-kick, Lloris saved point-blank but Simeon Jackson was there to smash home the rebound.
There was still time for another dramatic twist as Walker burst into the box and was fouled by Tierney.
Dempsey placed the ball down and fired to Bunn's left but the goalkeeper guessed correctly to turn the spot-kick away.
Norwich (4-4-1-1): Bunn; Whittaker, R Bennett, Turner, Tierney; Snodgrass, Fox (Tettey, 69), Howson, Surman (Holt, 82); Butterfield (Jackson, 69); Morison. Unused subs: Rudd, E Bennett, Garrido, Barnett.
Spurs (4-2-3-1): Lloris; Walker (Defoe, 89), Caulker, Dawson, Naughton; Carroll (Vertonghen, 79), Livermore (Huddlestone, 46); Falque, Sigurdsson, Bale; Dempsey. Unused subs: Cudicini, Smith, Lennon, Townsend.
Goals: Norwich - Tettey/Vertonghen own goal (83), Jackson (86). Spurs - Bale (66).
Yellow cards: Norwich - Bunn (88). Spurs - Bale (38).
Referee: Mr J Moss.