Andre Villas-Boas' men had laboured in search of a potentially lucrative winner after Stoke midfielder Charlie Adam was dismissed in the 47th minute for a second bookable offence.
But Adebayor tapped in a cross from Clint Dempsey from close range in the 83rd minute to take them back up to fourth and pile pressure on Arsene Wenger's men, who face Wigan on Tuesday night.
Spurs had started in dismal fashion as they fell behind to a Steven Nzonzi header after just three minutes, before Dempsey equalised with a brilliant piece of improvisation in the 20th minute, and from that point the visitors largely dominated.
Despite having nothing but pride to play for after edging out of the relegation dogfight, it was Stoke who started with more intent and Nzonzi headed an Adam free-kick past Hugo Lloris after Dempsey fouled Ryan Shotton on the right.
Dempsey had the ball in the net in the 15th minute but his effort was ruled out for offside, but five minutes later the American drew his side level with a superb piece of skill.
Stoke defender Marc Wilson and goalkeeper Asmir Begovic got in a muddle over a crossfield ball by Scott Parker, Wilson clearing the ball to Dempsey who responded by lobbing it straight back over the stranded Begovic from 30 yards.
The equaliser did not cause Stoke to lose any of their conviction, with a neat Adam chip into the box almost finding former Spurs striker Crouch, with Lloris reacting well to paw away the danger.
But Spurs began to get more of a grip on the game and almost grabbed a second five minutes from the break when Michael Dawson met a Gareth Bale cross with an angled header which was parried away by Begovic.
And Begovic came to Stoke's rescue again just before the break when he palmed away a low effort from the increasingly influential Bale in the box.
But Stoke also had a chance to go in ahead at the break after Adam brushed past Dempsey by the corner flag and dribbled a ball across goal which was eventually booted clear by Dawson.
Stoke's task got a lot tougher two minutes into the second half when Adam was dismissed by referee Kevin Friend after a clumsy challenge on Jan Vertonghen on the Spurs left.
The visitors almost capitalised immediately, a low corner striking Dempsey and deflecting just over Begovic's bar.
And they proceeded to pin Stoke back for long periods, Ryan Shawcross twice getting in vital interceptions to deny first Aaron Lennon then Bale in front of goal as a second Spurs goal began to look inevitable.
Shawcross was at his best again to end a clever run into the box from Tom Huddlestone, and when the ball sprung clear to Bale in the left side of the box, his low shot was well saved low by Begovic.
With a second goal continuing to elude them, tension began to seep into Spurs' play and Robert Huth almost took advantage, heading a Dean Whitehead free-kick just over after a foul by Kyle Walker on the left.
But Spurs finally grabbed their second seven minutes from time when Dempsey picked up a half-clearance and dribbled the ball across the six-yard box where Adebayor converted the simplest of chances.
Stoke came close to grabbing a shock equaliser deep into injury time when substitutes Kenwyne Jones and Geoff Cameron combined with the latter heading into the side-netting from a tight angle.