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.
It was Stoke who started with more intent and Nzonzi headed an Adam free-kick past Hugo Lloris in just the third minute.
Dempsey drew his side level with a superb piece of skill in the 20th minute.
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.
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.
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.
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.