Gasly, who started 10th, managed to finish the race in first position, retaining the lead after restarting the race, stopped by red flags after a serious accident involving Ferrari driver Charles Leclerc.
Spaniard Carlos Sainz from McLaren crossed the finish line second.
Top three were Canadian Lance Stroll from Racing Point.
Championship leader Mercedes driver Lewis Hamilton finished only seventh after serving a 10-second stop-and-go penalty for pits when it was prohibited.
Gasli's teammate, Russian Daniil Kvyat, finished the race in ninth place.
Earlier it was reported that Hamilton and Vettel were late for the kneeling ceremony at the Italian Grand Prix.