Elliott Bennett powered the ball into the far corner of Brad Friedel's net in the 66th minute from outside the penalty area for the winning goal.
The strike calmed Norwich manager Paul Lambert, who had been furious that two potential penalties were not awarded.
The first spot kick appeal was waved away just before Jermain Defoe completed a blistering counterattack to equalize in the 33rd.
Anthony Pilkington had put Norwich in front in the 13th after seizing on a mix-up in Tottenham's disorganized defense.
While Spurs remained in the fourth Champions League spot, they can be dislodged on Monday night if Chelsea wins at Fulham.
Newcastle is also a growing threat to Tottenham's hopes of returning to Europe's lucrative competition after its debut in the 2009-10 season. The northeast club beat Bolton 2-0 to join Tottenham on 59 points but in fifth place due to an inferior goal difference.
Tottenham's fortunes have rapidly plummeted since enjoying a 10-point lead in third over fourth-place Arsenal at the start of March.
Harry Redknapp's side has now won just one of its past eight league games, with the latest collapse starting after a breakdown in communication in the back four to gift Norwich its opening goal.
Aaron Wilbraham's shot was blocked by Younes Kaboul but Kyle Walker fluffed his attempt to clear the danger, the ball ricocheting off Tottenham defender Ledley King and Pilkington pouncing to fire it past goalkeeper Brad Friedel.
Tottenham was relying on Bale's powerful runs down the left to rip Norwich apart, but his 24th-minute shot was saved and Defoe failed to connect with his cross on another occasion.
The first penalty claim came when Norwich striker Grant Holt appeared to be shoved by King, prompting his manager to charge from the bench to near the tunnel to complain to fourth official Phil Dowd.
As the official was being admonished by Lambert, Tottenham launched a swift counterattack that produced the equalizer. Defoe raced onto Jake Livermore's pass and lifted the ball over advancing goalkeeper John Ruddy.
Norwich came close to regaining the lead, but Pilkington's effort was blocked by Friedel throwing himself at the ball.
Still incensed at the end of the first half, Lambert waited by for referee Michael Oliver to leave the pitch to confront him.
Norwich was denied another penalty on the hour when Emmanuel Adebayor's push on Wilbraham went unpunished.
The visitors were then rescued by the crossbar when Bale's shot from the right came off the underside before Bennett's winner.
