Substitute Gomis struck five minutes from time, meeting Jefferson Montero's cross with a header that crossed the line before it was scooped out by keeper David Ospina, the goal confirmed almost immediately to the referee by the goalline technology system.
Arsenal, who have three games left to play, remain on 70 points, three points behind City who have two games left and sit in pole position for automatic entry to next season's Champions League group stage.
Swansea goalkeeper Lukasz Fabianski frustrated his former side, the Poland international making several good saves in the second half to thwart the home side.
Arsenal, unchanged for fifth successive game, had rarely tested Fabianski in a tepid opening half.
It took almost an hour for Arsenal to muster their first shot on target with Olivier Giroud shooting straight at Fabianski.
Arsenal pressed forward with greater urgency and Fabianski denied Alexis Sanchez and then Theo Walcott with a fine double save.
(Writing by Justin Palmer; editing by Pritha Sarkar)
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