The Togo striker fired home his seventh goal of the season with a fine second-half finish at White Hart Lane on Sunday, a strike that ensured Tottenham replaced the Merseyside outfit in fifth place.
Everton had the better of much of the game but lacked a cutting edge in the absence of a recognised striker as their hopes of securing a top-four finish suffered a setback.
Tottenham were thrashed 5-1 by Manchester City in their last home game, but Tim Sherwood's side got back on track to move two points above Everton and three behind fourth-placed Liverpool.
Roberto Martinez's team's away form is becoming their downfall in the race for UEFA Champions League football, as they have won only one of their last six top-flight matches on their travels.
Fit-again midfielder Moussa Dembele replaced Roberto Soldado in the Tottenham starting line-up, while Andros Townsend was among the substitutes after recovering from a hamstring problem that has sidelined the winger since December.
Everton defender Seamus Coleman also returned from a hamstring injury and came into the side along with Steven Pienaar and Steven Naismith as Ross Barkley, John Stones and Aiden McGeady dropped to the bench.
The visitors lined up without a recognised striker as Naismith got the nod to play in a more advanced role, but they still showed plenty of attacking threat early on.
Leon Osman had four chances to open the scoring in the first eight minutes, twice shooting straight at Hugo Lloris and also forcing the France goalkeeper into an outstanding fingertip save with a right-foot strike before heading Naismith's corner just wide.
Despite Everton's bright start, they were almost caught out when Danny Rose was allowed too much space to foray forward and deliver a dangerous cross that only just evaded Adebayor.
Martinez’s side undoubtedly had the better of the first half, though, and the lively Kevin Mirallas embarked on an incisive run from inside his own half and burst into the penalty area before firing over the bar eight minutes before the break.
Mirallas was off target again five minutes into the second half with a right-foot drive from outside the area as Tottenham struggled to live with Everton's brisk passing and movement.
Townsend replaced Christian Eriksen after 58 minutes as Sherwood attempted to give the home side some impetus, and the winger's England team-mate Barkley replaced former Tottenham midfielder Pienaar for Everton five minutes later.
And it was Tottenham who broke the deadlock out of the blue after 64 minutes, when Kyle Walker picked out Adebayor with a free-kick that caught Everton out and the striker controlled the ball on his chest before beating Tim Howard with a clinical left-foot finish.
Gerard Deulofeu made his return from injury and Martinez also brought on McGeady in search of an equaliser, but Adebayor came close to sealing the points with a second goal when he fired over.
Jermain Defoe was given a run out in front of the White Hart Lane faithful for the final time before his move to Toronto FC and almost signed off in style with a goal in stoppage time, but Howard denied him.
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