Clint Dempsey levelled Steven Nzonzi's third-minute header with a fine strike in the 20th minute - firing into an empty net from fully 35 yards - and after dominating possession following Charlie Adam's sending off at the start of the second half, Adebayor clinched the win seven minutes from time.
It moves us back over Arsenal into fourth place in the Premier League with the Gunners now taking on FA Cup winners Wigan at the Emirates on Tuesday night.
Andre Villas-Boas made two changes from Wednesday night's 2-2 draw at Chelsea, with Steven Caulker and Dempsey taking over from Benoit Assou-Ekotto and Lewis Holtby. Jan Vertonghen switched to left-back, with Dempsey playing ahead of him off the left flank.
The home crowd were decked out in red and white t-shirts, providing a colourful backdrop to the game to mark Stoke City's 150th year. There was cause for home celebrations anyway with their Premier League status assured for another season, but these were upped considerably in the third minute when an Adam free-kick arrowed into the area and found the head of Nzonzi, who steered a powerful header past Hugo Lloris to put his side in the driving seat.
Gareth Bale fizzed in a shot two minutes later that Asmir Begovic was forced to tip to safety before shooting over 11 minutes in after being teed up by Dempsey.
A foothold in the game was slowly being established and a leveller was denied on 16 minutes when Dempsey had the ball in the net, but the assistant's flag was held aloft and celebrations were swiftly curtailed.
An equaliser duly arrived - very much deserved - on 20 minutes when Scott Parker teased the ball in the direction of a sprinting Aaron Lennon. Such was the panic this threat caused that both goalkeeper and Marc Wilson were drawn to the ball, with the defender intercepting, but the ball fell to Dempsey, who lofted over keeper and defenders and happily rattled the net.
It was a show of good character from the team in regrouping after the early setback and not allowing Stoke to gather any momentum after seizing such a swift advantage. Adebayor was once again a mobile and powerful presence in attack, dropping deep too in order to join up the play.
The striker stretched the Stoke defences on 35 minutes, dragging them across the area before dispatching a shot that he was certain struck the arm of Robert Huth. Begovic then made a stunning save to deny a Michael Dawson header from a Bale free-kick whipped in from the right.
Begovic foiled Bale after a speedy run just before half-time as tempers were getting a little frayed. Adam was guilty of a lunge at Vertonghen when the Belgian was thwarting a dangerous ball across the area and picked up a yellow card.
Just after half-time came a red card that you felt was coming. Adam fouled Vertonghen for the second time and referee Mr Friend reached for his second yellow and then the red.
A low-struck corner shortly after from Bale was helped on by Steven Caulker and then struck Dempsey, but the ball flew narrowly over the bar.
The pressure was being applied in waves but Stoke stood firm as the ball was being passed around them as the team sought an opening. Begovic was sharp in grabbing a low Bale cross-shot after a mazy run from Huddlestone, while Vertonghen was popping up as a key threat down the left flank.
Vertonghen headed just over from a Huddlestone free-kick on 68 minutes before we saw the introduction of his countryman Mousa Dembele, in place of Parker.
Dempsey let fly on 71 minutes with Begovic again tipping away for a corner. It was soon Defoe-time and the little striker came on in the 74th minute, taking over from Lennon to provide additional firepower.
The next big chance fell to Bale though, having been played in by Dembele, twisted and turned and flashed a shot just wide, with Dempsey free in a fair bit of space.
This was a performance that deserved reward, but it was Huth who came close with a header 10 minutes from time and the introduction of Gylfi Sigurdsson was looming.
Before he got the chance to come on, the lead was taken on 83 minutes, when a Bale shot deflected an fell at the feet of Dempsey in the area. The American remained composed and slid across goal to where Adebayor was waiting and he shot home with glee - and then a dance!
There were no more scares - and the points were safe.
Stoke City (4-4-2): Begovic; Shotton (Cameron, 74), Huth, Shawcross, Wilson; Adam, Nzonzi, Whitehead, Etherington (Wilkinson, 57); Crouch (Jerome, 57), Walters. Unused subs: Sorensen, Jones, Owen, Kightly.
Spurs (4-4-2): Lloris; Walker, Caulker, Dawson, Vertonghen; Lennon (Defoe, 74), Huddlestone, Parker (Dembele, 68), Dempsey (Sigurdsson, 84); Adebayor, Bale. Unused subs: Friedel, Naughton, Carroll, Holtby.
Yellow cards: Stoke - Huth, Begovic, Adam, Shotton. Spurs - Huddlestone.
Red card: Stoke - Adam (second yellow card).
Goals: Stoke - Nzonzi (3). Spurs - Dempsey (20), Adebayor (83).
Refere: Mr K Friend.
The Togo striker was in the right place at the right time to tap home from Clint Dempsey's cross as we recovered from Stoke's early goal to have the better of it at the Britannia Stadium.