The hosts had the better of the opening stages as Jonjo Shelvey forced Hugo Lloris into a smart stop from range before Wilfried Bony lashed a venomous strike against the bar with the Spurs goalkeeper well beaten.
However, it was Tim Sherwood's side who eventually opened the scoring as Adebayor headed home from close range after a pin-point delivery from Christian Eriksen who had combined well down the right with Aaron Lennon.
Spurs then doubled their lead just after the break when Chico Flores diverted Kyle Walker's cross into his own net after some patient build-up play from the visitors.
Adebayor claimed his second of the afternoon after Danny Rose had found him in space on the counter attack and the striker ensured that his rich vein of form continued as he calmly slotted past Gerhard Tremmel and into the bottom corner.
Bony pulled one back for the hosts as he expertly guided home Roland Lamah's scuffed cross from the edge of the area, but Spurs held on to comfortably claim all three points.
The win moves Tottenham up to fifth, level on points with fourth-placed Liverpool as Sherwood, who opted against sticking with a 4-4-2 formation here, remained unbeaten in the Premier League since taking over from Andre Villas-Boas.
Swansea, meanwhile, remain just three points above the relegation zone and their lengthy injury list and the alarming manner in which they faded following Adebayor's opener will be very troubling for manager Michael Laudrup.
Swansea had entered the game without a win in seven league games and their predicament had even led the usually reserved Laudrup to emit a rallying call that his side had six weeks to turn around their fortunes or face up to being involved in a relegation dogfight.
It appeared to have the desired effect as his players opened strongly as Shelvey was the first to work Lloris, turning away from Vlad Chiriches to unleash a fierce drive, before Bony fired a couple of yards wide on the turn.
Bony was causing a number of problems and he checked inside Michael Dawson and produced a rocket of a shot which beat Lloris but smashed against the crossbar.
Dawson was fortunate not to concede a penalty from a corner seconds later when he appeared to shove Bony, and Swansea's frustrations only grew when they fell behind after 35 minutes.
Eriksen produced a wonderful cross from the right to catch the Swansea defence out of position, with Adebayor completing the formalities from close range.
Nacer Chadli should have doubled Spurs' lead moments later, firing over after being left in acres of space.
Tremmel then punched away a stinging Adebayor volley and Swansea were ultimately relieved to hear the half-time whistle.
But matters did not improve for them after the break as Shelvey limped off with a hamstring problem to be replaced by Lamah.
Within minutes Tottenham doubled their lead. Lamah lost Walker completely and it proved costly as Flores could only turn the full-back's cross into his own net.
Lloris again had to be alert to block another powerful Bony effort, while Flores headed over from a corner as the hosts sought a way back into the game.
But Swansea's need to push forward left them exposed defensively.
Mousa Dembele missed a sitter after Lennon's pass outwitted the Swansea offside trap, but there was no such reprieve when Danny Rose raced down the left and pulled the ball across the box for Adebayor to calmly beat Tremmel and claim his second on the day and sixth in eight games.
Bony pulled one back for Swansea as he steered Lamah's scuffed cross beyond Lloris, but it was too little, too late for the hosts.