The visitors bossed the first half, with Hugo Lloris called into action on a number of occasions, but an improved display after the break from Spurs secured the three points.
The Togo striker fired home his seventh goal of the season with a fine second-half finish at White Hart Lane on Sunday to move just three points behind fourth-place Liverpool.
Everton had the better of much of the game but lacked a cutting edge in the absence of a recognized striker as its hopes of securing a top-four finish suffered a setback.
Tottenham was thrashed 5-1 by Manchester City in its last home game, but Tim Sherwood's side got back on track to move two points above Everton.
Everton's away form is becoming its downfall in the race for a UEFA Champions League place, as it has won only one of its last six top-flight matches on the road.
Fit-again midfielder Moussa Dembele replaced Roberto Soldado in the Tottenham starting lineup, 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 recognized 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-footed strike before heading Naismith's corner just wide.
Despite Everton's bright start, it was 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-footed 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 teammate Barkley replaced Pienaar for Everton five minutes later.
And it was Tottenham that broke the deadlock out of the blue after 64 minutes, when Adebayor was picked out with a free kick that caught Everton off guard and the striker controlled the ball on his chest before beating Tim Howard with a clinical left-footed finish.
Gerard Deulofeu made his return from injury and Martinez also brought on McGeady in search of an equalizer, 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.