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-placed Liverpool.
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.
|VIEW FROM WHITE HART LANE
|By Greg Stobart
Tim Sherwood's clenched fists and relieved look to the heavens told the story of a victory Tottenham did not deserve today.
But the Londoners are three points off fourth-placed Liverpool and in Emmanuel Adebayor have a striker who can win matches single-handedly.
Here, the Togolese forward scored the only goal of the game with a fine finish in the second half, the only memorable opportunity of the whole 90 minutes for the hosts.
Everton dominated long periods of the game, made plenty of chances and looked comfortable until Adebayor’s intervention - but the visitors badly missed the attacking presence of Romelu Lukaku.
With the Belgian injured, Steven Naismith deputised as the attacking spearhead and, ultimately, it was the difference between the two teams' strikers that decided this game.
There was also a nice touch at the end as many Spurs fans stayed behind to applaud Jermain Defoe on a lap of honour after his last home appearance before he leaves for Toronto FC at the end of February.
Roberto Martinez's team's away form is becoming their downfall in the race for 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.