The Togo striker, who was frozen out of the side under former coach Andre Villas-Boas, took his tally for the season to nine in all competitions as Spurs won on the road for the ninth Premier League game this season.
Adebayor, who only managed eight goals in the whole of last season, scored an early tap-in and added the third late on after Paulinho had doubled the visitors' lead.
Nacer Chadli completed the rout with a superb strike two minutes from time to hand Newcastle a fourth straight home league defeat, with Alan Pardew's men having drawn a blank in all of those matches.
The victory keeps Tim Sherwood's side in the hunt for fourth spot as Spurs seek a return to Europe's premier club competition.
It was Tottenham who started brightest and after four minutes Moussa Dembele muscled past Davide Santon on the right to find Adebayor in the box but his shot was deflected for a corner.
Papiss Cisse, making his first start in four months for the home side, should have opened the scoring inside the opening eight minutes but fired his close-range strike straight at Hugo Loris.
Adebayor broke the deadlock 19 minutes in with a simple finish from six yards after Tim Krul spilled Nabil Bentaleb's cross from the left.
Sammy Ameobi tried to force an equaliser with a great run and shot through the middle but strong defending from Younes Kaboul stopped him in his tracks.
Tottenham almost increased their advantage when Kyle Walker whipped a dangerous ball in from the right but the Newcastle defence crowded Aaron Lennon out of the box.
Lennon almost scored Tottenham's second before the break when he cut in from the right and unleashed a shot that beat Krul but hit the bottom of his right-hand post.
Seven minutes after the restart Paulinho followed up a saved Adebayor shot to convert Tottenham's second and underline their dominance.
Newcastle introduced Hatem Ben Arfa from the bench and started to get a grip on the game after 65 minutes as Paul Dummett fizzed a shot just past Lloris' goal.
But Adebayor clinched his second of the game when he knocked his shot into the ground and over the head of Krul with eight minutes left.
Chadli saved the best until last, curling home from distance to cap an excellent display from Sherwood's charges.
