It was a fourth straight league victory for Harry Redknapp's men, but they were made to fight every inch of the way by a City side who had Dean Whitehead sent off during a disastrous start to the second half.
Jermain Defoe was sidelined with a hamstring injury, so Redknapp teamed up giants Peter Crouch and Roman Pavlyuchenko in attack.
There was no place for Stoke's Ryan Shawcross, despite his availability following a three-match ban in the wake of the tackle which broke Aaron Ramsey's leg.
Stoke made most of the early running, but it was Spurs who had the first serious dig at goal when the ball was played to Niko Kranjcar in the 18th minute.
The Croatian drove a 20-yard shot which was too hot for Thomas Sorensen to hold and the City keeper pounced on the rebound as Crouch followed in.
Stoke hit back and Abdoulaye Faye should have done better than head wide from eight yards after Dave Kitson had flicked on Glenn Whelan's cross.
And Kitson's header from a Rory Delap long throw had Heurelho Gomes scampering to save, before a cross from the lively Kitson caused momentary panic in the visiting defence.
Pavlyuchenko was replaced by Gudjohnsen after limping off in the 36th minute and a high-paced first half ended with Faye denying Gareth Bale with a magnificent tackle and Gomes saving from Ricardo Fuller.
But the game swung Tottenham's way big style in the first four minutes of the second half - starting with Gudjohnsen's first goal for the club just 22 seconds after the resumption.
Crouch played a perfect pass in behind Faye and the Icelander outpaced the defender to strike an unstoppable half-volley into the roof of the net from ten yards.
Stunned Stoke were then reduced to ten men in the 49th minute when Whitehead made the long walk after picking up a second yellow card for a foul on Luka Modric.
Tottenham, with Modric and Kranjcar revelling in the extra space, went for the jugular and the home defence was creaking under a constant bombardment.
But Stoke gained a 64th minute lifeline during a rare attack when Benoit Assou-Ekotto unnecessarily flattened Kitson from a Danny Collins cross.
Former Spurs man Matty Etherington sent Gomes the wrong way from the spot and Stoke should have gone ahead shortly afterwards when Fuller missed a sitter.
The Jamaican lifted a left-foot shot hopelessly over the bar from six yards after a miscued shot from Danny Higginbotham had drifted into his path.
Tottenham made them pay when Assou-Ekotto atoned for his error by playing a big part in his team's 77th minute winner.
The left-back bombed forward to roll an inviting cross for Gudjohnsen to cleverly step over, before the onrushing Kranjcar hammered an unstoppable rising shot past Sorensen.
Stoke were still far from finished, however, and Spurs were hanging on at the death as Fuller and substitute Mamady Sidibe, who fell over with the goal at his mercy, went close to a second equaliser.