Spurs started the brighter and were gifted two early chances courtesy of poor defending from Davide Santon.
First, Moussa Dembele caught the Italian in possession and when he squared to Adebayor his shot was deflected narrowly wide.
Then, moments later, Santon allowed Paulinho a sight of goal and the former Corinthians man tested Tim Krul with a stinging drive from close range.
Newcastle briefly threatened and Papiss Cisse really should have done better with Luuk de Jong's knock down but the Senegal striker's shot was turned away well by Hugo Lloris.
On 19 minutes, though, Tim Sherwood's side got the goal their play deserved. Nabil Bentaleb created some space down the left and when Krul could only palm his delivery out Adebayor was on hand to tap in from six yards.
The visitors then nearly doubled their advantage minutes later when Kyle Walker fizzed a ball across the Newcastle penalty area but Mike Williamson somehow scrambled it away before Aaron Lennon could convert.
Alan Pardew's side improved after that and probably should have equalised through Cisse, but his lob over Lloris lacked power and direction and drifted wide.
After the break, the Gallowgate End thought they might see a flurry of attacks in the direction of Lloris' goal but it was Tottenham who doubled their advantage and Krul again should probably have done better.
Adebayor had a shot from just outside the area which the Dutch 'keeper couldn't hold and Paulinho reacted quickest to slot into the net.
Paul Dummett, who was introduced at half-time in place of the disappointing Santon, continued to press down the left but his final ball lacked the quality his build-up play deserved.
Hatem Ben Arfa was also introduced in the hope of giving his side and the home support a lift and his cross to Mathieu Debuchy was somehow tipped over the bar by the exceptional Lloris.
Sherwood had strength in depth from the bench too and when Andros Townsend's shot was saved, Adebayor's follow-up looped over Krul and into the corner.
With two minutes remaining the rout was completed when Nacer Chadli found the top corner beautifully from the edge of the area to rubber stamp an impressive away performance.
Townsend could have heaped more misery on Pardew's side but Krul raced out to deny him from point-blank range.
The result keeps alive Tottenham's hopes of playing Champions League football while Newcastle look set to miss out on Europe once again.