The hosts had led through Steven N'Zonzi's early header and despite Clint Dempsey's equaliser (20) they looked to be holding on to a point having survived for almost all of the second half without the red-carded Charlie Adam.
But up-popped striker Adebayor with seven minutes left to slot in pulled-back cross from the lively Clint Dempsey and secure the spoils to hoist them into the top four for the first time in a month, leading Arsenal by two points having played a game more.
And so the Premier League drama rumbles on with Spurs facing Sunderland next week in their last game, while the Gunners meet FA Cup winners Wigan on Tuesday, followed by a trip to Newcastle.
It had started so well for the hosts. Their 150th anniversary and recently pulling away from the drop zone with two wins and a draw in their last three games were reasons enough to celebrate for the home fans, but they wouldn't have expected such an early gift.
The clock read 150 seconds when Adam floated in a free-kick for N'Zonzi to rise unmarked and nod home with keeper Hugo Lloris only able to palm it in off the upright.
Apart from a long-range bullet from Gareth Bale tipped over comfortably by Asmir Begovic, the hosts were energetic and defending resolutely.
But poor positioning by the Bosnian keeper on 19 minutes handed their visitors a lifeline. He strolled to the edge of the box after Scott Parker's through-ball and looked on in horror as Mark Wilson's clearance fell to Dempsey who improvised brilliantly to slide the ball in from 40 yards.
Chances were few and far between but as the interval approached, Begovic was called into action, making a smart pushed save from Michael Dawson's header after a cunning Bale cross.
Within 65 seconds of the restart, Stoke's cause was given a harsh but avoidable slap with Adam given his marching orders for a second clumsy tackle and a second yellow card with Jan Vertonghen on the wrong end of a sliding challenge.
Sensing their moment to punish, the London visitors poured forward and from a Bale corner, the ball was shinned over by Dempsey.
Stoke's mission swiftly evolved into a salvage mission as boss Tony Pulis introduced Andy Wilkinson and Cameron Jerome in place of Peter Crouch and Matthew Etherington.
A wayward Vertonghen header and a 30-yard blast from Dempsey caused a few flutters for Stoke's battlers and they would have been relieved (75) to see Bale turn inside the box and curl inches wide.
Robert Huth nodded wide in a rare attack for Stoke, but after scoring goals beyond the 80th minute in their six previous matches, Spurs were at it again with Adebayor's late strike as they rejuvenated their hopes of qualifying for Champions League football.