Rafael van der Vaart had earned Spurs a half-time lead with a fine finish at the end of a flowing move in the 19th minute, but Gylfi Sigurdsson drew Swansea level with an equally good and equally deserved goal just before the hour mark.
However, Adebayor headed the hosts back ahead in the 73rd minute and then sealed the victory with another header four minutes from time as Harry Redknapp's side moved level on points with third-placed Arsenal with their first Premier League win in seven weeks.
Despite their recent woes, Spurs started confidently on a sunny afternoon in north London.
Luka Modric skipped past two Swansea players and found Van der Vaart, but the Dutchman's shot went just wide courtesy of a huge deflection off Neil Taylor.
Van der Vaart then clumsily misplaced a pass while clearing from his own box and the ball found its way to Wayne Routledge, who screwed his shot just wide.
Spurs came forward in the 12th minute through Gareth Bale, who raced down the left flank, but Adebayor could not convert his cross.
Taylor gave the home fans a scare after 15 minutes when he wriggled past William Gallas to sneak into the box but Friedel saved with his legs.
The home fans' nerves eased in the 19th minute, however, when Spurs took the lead from a lightning counter-attack.
Modric released Bale down the left and Ashley Williams cleared his cross straight into the path of Van der Vaart, who finished well from the edge of the area.
Michel Vorm pulled off a fantastic diving save to deny Younes Kaboul moments later as Spurs sought a second goal.
Swansea still looked threatening on the attack, though, and would have equalised had Williams not headed over Sigurdsson's corner.
Bale went close on two occasions and Modric drove just wide in the dying minutes of the first half.
Friedel produced a world-class save to deny Sigurdsson in the 50th minute, the American moving across his line before athletically tipping the forward's screaming 25-yard shot wide.
Sigurdsson soon had his revenge over Friedel by beating the 40-year-old to give Swansea a shock equaliser just before the hour.
Joe Allen's deflected shot fell in to the path of Routledge, who held off Benoit Assou-Ekotto's challenge and chested the ball down to Sigurdsson. The Iceland striker fired a volley into the ground that looped over Friedel from 18 yards.
Spurs looked for an immediate response but they were denied by a top-drawer save from Vorm. Van der Vaart rose to meet Bale's corner at the near put but the forward's fellow Dutchman tipped his header around his near post.
Redknapp came to the touchline for practically the first time in the game, clearly concerned his side were about to throw away the chance to move level with Arsenal.
Bale responded to his manager's presence by firing a low shot that Vorm did well to save.
Redknapp brought Lennon into the fray for Sandro with 20 minutes to go while Nathan Dyer replaced Routledge for the Swans.
Kaboul surged forward and found Adebayor on the edge of the box but his shot was weak.
The Togo striker soon made amends, however, as he put Spurs back in front with 18 minutes left. The 28-year-old rose majestically in the centre of the box to beat his marker and power home a head from Van der Vaart's corner before embarking on a wild celebration in front of the home fans.
Bale went down in the box but saw his penalty appeal waved away while Swans boss Brendan Rodgers replaced Danny Graham with Luke Moore.
Sigurdsson fired a warning shot to the hosts when he curled a delightful free-kick just wide of Vorm's goal, before Adebayor sealed the contest with four minutes left when he nodded home Lennon's cross.