The two top-four chasers played out a goalless draw at Goodison Park in early November but on this occasion, Spurs' Togo international frontman was able to make the breakthrough when he brought down Kyle Walker's chipped free-kick in the 65th minute and fired past Tim Howard.
Everton looked to have a strong penalty claim when Spurs substitute Etienne Capoue caught Seamus Coleman in stoppage time, but referee Mark Clattenburg was unmoved.
The victory at White Hart Lane moved Tim Sherwood's side above the Toffees into fifth spot, three points behind Liverpool.
Andros Townsend returned to the Spurs squad after seven weeks out. The England winger suffered a hamstring injury against West Ham on December 18 and was named among the substitutes, while Mousa Dembele replaced Roberto Soldado in the starting line-up.
Everton made three changes after beating Aston Villa, bringing in Coleman, Steven Pienaar and Steven Naismith for John Stones, Aiden McGeady and Ross Barkley, who all dropped to the bench alongside the fit-again Gerard Deulofeu and Lacina Traore.
Few would have expected a home win after the way Everton flew out of the blocks, with Leon Osman coming close four times inside the opening eight minutes.
The midfielder sent a warning shot across the bows with a curling strike shortly after kick-off and followed it up with a fine half-volley, which was destined for the net until a superb one-handed save by Hugo Lloris.
The England international headed over from the resulting corner and continued to prove a nuisance, although he wasted the easiest effort of the lot by firing straight at Lloris after neat work by the returning Steven Naismith.
Sherwood was understandably agitated by Spurs' sloppy start, with a Dembele half-chance all they had to show for their endeavours until Christian Eriksen's long-range free-kick dipped onto the roof of Howard's goal.
Lloris, unaware the offside flag had been raised, produced another wonderful save to stop Kevin Mirallas converting a Leighton Baines cross, before the Spurs goalkeeper swept up after confusion at the back and Coleman had a tame effort saved.
Spurs could have taken an undeserved lead when a low Danny Rose cross just evaded Adebayor. Kevin Mirallas then slalomed through on goal before firing over as the sides went in level at the break.
Townsend replaced Eriksen after 59 minutes in a bid to add a new dimension to the Spurs attack. The winger had barely had time to settle into proceedings when Spurs caught Everton napping to break the deadlock through Adebayor's astute finish.
Walker lofted a quickly-taken free-kick up to the edge of the box where Adebayor held off Coleman before drilling the ball low past Howard. It was Spurs' first shot on target.
The relief inside White Hart Lane was palpable, especially as a timely Michael Dawson block denied Everton minutes earlier.
Spurs were now looking composed, though, and Adebayor flashed a strike over as they looked to put the game out of reach.
Everton were not giving up and continued to press, with Lloris called into action to thwart a deflected cross from substitute McGeady.
The onslaught continued and they had a strong stoppage-time penalty appeal waved away after substitute Capoue brought down Coleman as Spurs held out for the win - the ideal send-off for Jermain Defoe, who came on with six minutes remaining for his final league appearance at White Hart Lane before joining Toronto.
Both clubs are back in league action on Wednesday night. Everton host Crystal Palace, while Spurs travel north to face Newcastle.