In an entertaining and at times controversial contest at White Hart Lane, Antony Pilkington put Norwich in front after 13 minutes and the visiting side should have then had a penalty when Ledley King brought down Grant Holt, only for referee Michael Oliver to infuriate Paul Lambert when allowing play to continue.
Norwich were punished inside 60 seconds when Jermain Defoe lifted the ball over John Ruddy but responded in a thrilling second half when Elliott Bennett scored with a wonderful effort after Gareth Bale had hit the bar for Spurs.
The result, combined with Newcastle's win at home to Bolton, means Spurs have been caught by Alan Pardew's side with both clubs now on 59 points with five games remaining, two behind third-placed Arsenal who also have a game in hand.
Norwich, meanwhile, move into ninth place after becoming only the third team following the Manchester clubs to win at White Hart Lane this season.
Tottenham made five changes from the side that drew 0-0 at Sunderland on Saturday, one of which was enforced as Scott Parker missed out due to injury. They also reverted to a 4-4-2 formation, matched by Norwich, and looked somewhat uncomfortable in a difficult first half for the home side.
Though it was Tottenham who enjoyed the first real chance of the game when Defoe's firm low drive caused Ruddy to spill the ball, the opening 45 minutes were characterised by some uncertain defending from Spurs, who in theory should have been strengthened by the return of King.
Instead the centre-back was given a torrid time by Holt as the striker's presence unsettled the home defence time and again. Though Holt was not involved in the opening goal after 13 minutes, it was a moment symptomatic of Spurs' problems at the back.
Younes Kaboul was caught in possession 20 yards from his own goal before both King and Kyle Walker contrived to fail to clear the ball and instead put it into the path of Pilkington, who produced a calm finish to slot the ball past Friedel.
A clumsy foul from Defoe allowed Bradley Johnson to fire in a vicious low free kick for Norwich, who were then caught out on the break as Bale burst forward down the left and delivered a ball that fizzed across the box. Defoe was only inches away from connecting.
A key moment in the game came after 32 minutes. Holt turned King all too easily in the box before being dragged to ground by the Spurs defender. It should have been a red card for King and a penalty for Norwich, yet referee Oliver waved play on, much to the disgust of the Norwich players and Lambert himself, who protested furiously on the sidelines.
Their anger was only aggravated inside a minute when Spurs drew level. Norwich were caught in possession in midfield and Jake Livermore played in Defoe with a perfectly-weighted through ball. The striker drew Ruddy off his line and then lifted the ball over the keeper with a delicate finish.
Norwich continued to unsettle Spurs with their application and industry - Elliott Ward testing Brad Friedel with a volley from a corner - yet they went in level at the break as referee Oliver was pursued by Lambert and his players, still disbelieving at his decision not to award a penalty.
Tottenham introduced top scorer Emmanuel Adebayor for Louis Saha at the break and the result was a far more cohesive performance in attack. Just minutes after the restart, Bale was denied by a fine block after connecting with a cut-back from Aaron Lennon.
After 53 minutes, Adebayor brought his presence to bear. Benoit Assou-Ekotto played a pass into the striker and, with his back to goal in the box, Adebayor returned the favour with a lovely flicked backheel. Assou-Ekotto raced onto the pass but saw his shot brilliantly saved by Ruddy in the Norwich goal.
Spurs continued to look vulnerable at the back though and were opened up after 58 minutes. Holt appeared to be just offside when racing onto a pass in the left channel but lifted a neat pass to Johnson at the back post. The midfielder had to deal with a bouncing ball but should have thumped the ball into the net instead of firing off target.
In a breathless, end-to-end second half, Norwich felt aggrieved once again just after the hour mark when Adebayor clumsily knocked Aaron Wilbraham to ground in the box but no penalty was awarded, before Spurs responded when Bale smacked the bar after checking back onto his left foot following a neat pass from Modric.
On 66 minutes, Norwich took the lead again. Martin played a pass infield to Elliott Bennett, and with Spurs standing off the midfielder advanced to the edge of the box before beating Friedel with a tremendous effort that flew across the goalkeeper and inside the far post.
As the chances continued to flow for both sides, Howson was guilty of a poor pass when he should have shot himself and Bale flashed a header just wide. Norwich were then appealing for a penalty for a third time when a cross from Steve Morison struck the arm of Nelsen, only to be turned down by referee Oliver once more.
But Oliver avoided the ire of Norwich players and staff as the visiting side held on for a famous victory that leaves Tottenham's aspirations of returning to the Champions League looking rather precarious.