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.
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.
