The striker connected with two fantastic headers to secure a vital three-point haul after Swansea had pegged back Rafa van der Vaart's opener via Gylfi Sigurdsson's strike on 59 minutes.
Adebayor was head and shoulders above the rest of the penalty area to power home van der Vaart's corner for 2-1 after 73 minutes.
He then repeated the trick after returning substitute Aaron Lennon chipped up an open invitation to score from the right byline four minutes from time.
Those were goals number 12 and 13 for 'Ade' in the Premier League this term and with the Gunners losing at QPR on Saturday, took us level on points with just one goal separating the teams in third and fourth.
Harry Redknapp made four changes to the team that beat Bolton in the FA Cup on Tuesday.
Brad Friedel was back in goal and we lined-up with a new central defence as Younes Kaboul and William Gallas took over from Ledley King and Ryan Nelsen. Sandro was back in central midfield for Jake Livermore with Lennon back on the bench after the best part of a month sidelined with a hamstring injury.
Gareth Bale was outstanding and involved in everything as we dominated the first half.
After near misses from van der Vaart and Adebayor, Bale was sent clear again by Luka Modric in the 19th minute and this time his low cross found van der Vaart, who finished sweetly from the edge of the box.
Michael Vorm turned away Younes Kaboul's header from van der Vaart's corner before Ahsley Williams headed wide Swansea's best chance of the half.
Bale was inches off target on two occasions before the final move of the half saw the impressive Benoit Assou-Ekotto win possession and Bale roll the ball into Modric, who drilled wide from 20 yards.
Swansea enjoyed their best spell at the start of the second half and Friedel was forced into one of his best saves of the season five minutes in, when he used all of his reach to tip Sigurdsson's 30-yarder around the post.
However, Sigurdsson did get the better of the American nine minutes later to draw the Welsh side level.
There was a stroke of fortune about the goal as Sandro's fine block on Joe Allen's shot fell into the path of Sigurdsson, who hammered his shot into the ground and it zipped up off the surface, over Friedel and into the corner.
We were soon back on top though and Vorm did well to palm away van der Vaart's glancing header from Bale's cross - almost a carbon-copy of his recent goal against Stoke - before fielding Bale's stinger from 25 yards.
Momentum was building again and Lennon entered the fray for Sandro with 20 minutes remaining.
Kaboul strode forward for fully 50 yards before laying off to Adebayor, who tried to place his effort but only hit Vorm.
Two minutes later though and the big goal. It was simplicity itself as well as van der Vaart pinged over a corner and Adebayor towered over everyone to plant his header home for 2-1.
Bale continued to drive forward and after beating numerous players found Modric, whose shot was blocked.
Modric was in again after a one-two with Lennon, but this time shot straight at Vorm.
The game just needed that final flourish to cap the afternoon and it arrived with four minutes left as Lennon delivered from the right byline and Adebayor headed home for 3-1.
Spurs (4-3-3): Friedel; Walker, Kaboul, Gallas, Assou-Ekotto (Rose, 87); Parker, Modric, Sandro (Lennon, 70); Van der Vaart (Livermore, 85), Bale; Adebayor. Unused subs: Cudicini, Nelsen, Defoe, Saha.
Swansea (4-2-3-1): Vorm; Rangel, Monk, Williams, Taylor; Allen (Gower, 92), Britton; Routledge (Dyer, 70), Sigurdsson, Sinclair; Graham (Moore, 79). Unused subs: Tremmel, Tate, McEachran, Lita.
Goals: Spurs - van der Vaart (19), Adebayor (73, 86). Swansea - Sigurdsson (59).
Referee: Mr A Marriner.