Michael Laudrup's men had the better of the opening exchanges, with Wilfried Bony smashing the crossbar from long range, but it was the visitors who took the lead through Adebayor, who powered home a devilish first-time cross from Christian Eriksen.
Spurs doubled their advantage shortly after the restart as Kyle Walker snuck down the right flank and drilled in a low cross that was turned into his own net by the unfortunate Chico Flores.
Tottenham continued to be a threat on the break and another swift counter saw Adebayor calmly bag his second of the afternoon, though Spurs were given a late scare when Bony cushioned a deft volley into the bottom corner — though it would be only a consolation for the Swans.
Laudrup, dealing with a host of injuries including the absence of Michu, made just one change from the side beaten by Manchester United last time out as Chico replaced Jose Canas, with Jordi Amat moving into midfield. Spurs also made one change from their last outing as Nacer Chadli came in for Roberto Soldado — the Spain international dropping to the bench.
Despite having won just twice at home in the Premier League this season, Swansea began brightly, with Jonjo Shelvey at the center of things for the hosts early on.
He had Swansea's first effort on target after 13 minutes, shrugging off the attentions of Vlad Chiriches before forcing Hugo Lloris into a smart stop with a long-range drive. Bony flashed a shot wide seven minutes later and the Ivorian then rattled the crossbar with a venomous strike from outside the penalty area.
Spurs' attack had looked out of sorts for much of the half but Adebayor found the net with their first real effort of the game. Eriksen's pinpoint cross was met by the former Arsenal striker, who registered his fifth goal in eight games from close range.
The goal sparked Spurs into life and Adebayor could have had a second six minutes before the break, only for Gerhard Tremmel to keep out his vicious volley.
An own goal from Chico double Spurs' advantage early in the second half, with the Spaniard diverting Walker's cross into his own net after Roland Lamah had failed to track the right back.
Adebayor then scored his second 19 minutes from time, calmly collecting Danny Rose's cross and rolling the ball past Tremmel just moments after Mousa Dembele had spurned a glorious chance to extend the lead.
That strike effectively wrapped up all three points to help Spurs to their fifth consecutive away win in the Premier League — the first time the north Londoners have managed such a run since the 1960-61 campaign.
Bony, a rare bright spark for the home side, eventually got on the score sheet with an expert volley after 78 minutes, though his 14th goal of the season made little difference to the result.
The loss means Laudrup's men remain just three points above the relegation zone ahead of clashes with Fulham, West Ham and Cardiff City — all of whom occupy spots in the bottom four — while Sherwood remains unbeaten in the league for Tottenham.