The striker headed Spurs into a 35th-minute lead and then tapped in their third after a Chico Flores own goal had doubled the visitors' lead.
Wilfried Bony hit the bar and had a good penalty appeal for a push by Michael Dawson turned down in the first half, before getting Swansea's consolation late on.
The win moves Tottenham up to fifth, level on points with fourth-placed Liverpool as Tim 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 of face up to being involved in a relegation dogfight.
It appeared to have the desired effect as his players opened strongly.
Jonjo Shelvey was the first to work Hugo Lloris as he turned away from Vlad Chiriches and unleashed a fierce drive, with Bony firing a couple of yards wide on the turn.
Swansea's desire to push forward led to occasional midfield lapses, but Spurs were having trouble handling their hosts with the excellent Bony in particular causing all manner of problems.
The Ivorian beautifully checked inside Dawson and produced a howitzer of a shot which beat Lloris but smashed against the crossbar.
Dawson was fortunate not to concede a penalty from a corner seconds later, the Tottenham captain clearly pushed Bony as he rose to try and get in a header but referee Martin Atkinson, who was looking straight at the incident, did not act.
And Swansea's frustrations only grew when they fell behind.
Christian Eriksen produced a wonderful cross from the right to catch the Swansea defence out of position, with Adebayor completing the formalities from close range.
The Welsh club were rocked and any confidence built from a good opening half-hour quickly evaporated.
Nacer Chadli should have doubled Spurs lead moments later, firing over after being left in acres of space.
Gerhard 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 as Shelvey limped off with a hamstring problem to be replaced by Roland Lamah.
The former Liverpool midfielder joins an injury list which already included Michu, Michel Vorm, Jonathan de Guzman, Pablo Hernandez, Nathan Dyer, Jose Canas and Garry Monk.
Within minutes Tottenham doubled their lead. Lamah lost Kyle 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.
Moussa Dembele missed one sitter afer 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.