The victory ended a long drought for the Italian Formula One team, whose last win was at the Spanish Grand Prix in 2013, and was the 40th of the German's career.
Starting from second place on the grid behind Lewis Hamilton's seemingly unbeatable Mercedes, Vettel and Ferrari produced a tactical masterclass to overhaul the Briton and cross the line 8.5 seconds ahead of the double world champion.
"Numero uno is back, Ferrari is back," Vettel was told over the team radio as he took the chequered flag. "Grazie, grazie. Forza Ferrari" he shouted in reply.
Hamilton's team mate Nico Rosberg finished a distant third while 17-year-old Max Verstappen claimed seventh place for Toro Rosso to become the youngest ever points scorer in Formula One history.
The race got off to a clean start with Hamilton pulling away from Vettel on the first lap, but the race turned in Ferrari's favour when Sauber's Marcus Ericsson spun off at the start of lap four to bring out the safety car.
While Mercedes pulled both their cars into the pits to put on hard tyres, Vettel stayed out on the medium compound and made the lead stick until the end of the race.
Vettel, who failed to win a race for Red Bull last year after four successive titles, has now won grands prix for three separate teams -- Toro Rosso, Red Bull and Ferrari.
(Reporting by John O'Brien; Editing by Peter Rutherford/Alan Baldwin)
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