Gomis, who has a history of fainting, fell to the ground in the centre circle after Nacer Chadli had given Spurs a seventh-minute lead.
Spurs players had their hands on their heads as the 29-year-old was tended to by medical staff and the incident brought back memories of Fabrice Muamba’s cardiac arrest at the same ground in 2012.
But the striker regained consciousness in time to acknowledge the crowd as he was stretchered off and Swansea confirmed afterwards their forward was “fine”.
Ki Sung-yueng equalised after the game restarted but Spurs were dominant in the second half and opened up a 3-1 lead thanks to goals from Ryan Mason and Andros Townsend.
Gylfi Sigurdsson reduced the deficit with a late volley but Mauricio Pochettino’s side held on to secure their first victory in almost a month thanks to a spectacular save in injury-time from Hugo Lloris.
The win is a welcome relief for Spurs after the recent disappointments of exiting the Europa League and losing the Capital One Cup final to Chelsea.
Pochettino stuck with the same starting line-up that were beaten at Wembley and they soon took the lead as Danny Rose whipped in a cross for Chadli to volley home.
But the goal was soon forgotten as Gomis collapsed, receiving medical attention for four minutes, before being taken off on a stretcher and with an oxygen mask.
When play resumed, Swansea were soon level, Sigurdsson hitting the woodwork with a brilliant free-kick and Ki alert to finish from a tight angle when the ball was played back into the box.
There were few chances in the remainder of an evenly-contested half but Spurs took control after the break.
Moments after Harry Kane went close with a header, Mason fired past Lukasz Fabianski for his first Premier League goal after Christian Eriksen’s mazy run cut through the Swansea defence.
Townsend put the hosts firmly in charge nine minutes later after a good advantage from referee Michael Oliver, who allowed play to continue following a foul on Nabil Bentaleb just outside the Spurs box.
Spurs countered at pace and Townsend raced down the left wing, cut inside a defender and fired into the far corner.
The home side looked in the mood for more and Kyle Walker’s deflected shot went narrowly wide of the post while substitute Roberto Soldado twice went close to adding a fourth.
But Spurs – who paid tribute to the late Dave Mackay with a minute’s applause before kick-off – could not put the game to bed and Swansea set up a tense finish when Sigurdsson volleyed home from Jefferson Montero’s cross in the 89th minute.
Swansea pushed for a late equaliser and only Lloris’ late heroics prevented the points being shared.
Tottenham: Lloris (7), Rose (7), Dier (5), Vertonghen (5), Walker (6), Bentaleb (6), Mason (7), Chadli (7), Eriksen (6), Mason (7), Townsend (7), Kane (5).
Subs: Davies (6), Dembele (6), Soldado (6).
Swansea: Fabianski (5), Taylor (5), Fernandez (5), Williams (5), Naughton (5), Cork (5), Ki (7), Shelvey (6), Routledge (6), Sigurdsson (7), Gomis (5).
Subs: Montero (6), Oliveira (5).
Man of the match: Andros Townsend.
Watch Wednesday’s goals on SSNHQ from midnight.