The result had been in doubt - even as the national anthem played on the podium, it emerged that Mercedes had been summoned to the stewards for a potential breach of the minimum starting tyre pressure regulations - but after an investigation the FIA confirmed that no further action would be taken, meaning Hamilton held on to his victory.
Ferrari's hopes of defeating the Silver Arrows took a knock as soon as the lights went out, as Kimi Raikkonen - alongside Hamilton on the front row of the grid - failed to get away cleanly and dropped to the back of the pack.
It was thus left to Vettel to give chase, but the German had no answer to Hamilton’s pace and the world champion went unchallenged throughout the 53 laps as he brought his Silver Arrow home 25 seconds clear.
The final podium position was the result of a super-tense intra-team Williams battle. Felipe Massa had run third until the pits stops, when both he and Valtteri Bottas were undercut by Rosberg, and the Brazilian was untroubled by the Finn until the closing laps when his rear tyres were wearing out. For the last three laps Bottas was locked onto Massa's tail, but his more experienced colleague proved unflappable and breathed a massive sigh of relief after reaching the chequered flag a mere 0.3s before his partner.
Raikkonen staged a great recovery to take fifth, having overtaken Sergio Perez's Force India with six laps to run. In the end Nico Hulkenberg just hung on to take seventh in the sister car, but that too was a mighty close-run thing after intense pressure from the Sauber of Marcus Ericsson.
The Swede, though, ended up falling victim to a charging Daniel Ricciardo on the final lap. The Red Bull driver was one of only four men to start on the medium tyre, allowing him to switch to softs at the end, and charge through to eighth from 19th on the grid. He finished just 0.5s behind Hulkenberg at the flag, with Ericsson another 0.2s adrift.
Daniil Kvyat also put the hammer down in the second Red Bull to climb from the back and took the final point in 10th. Fellow back-of-the-grid stable mates Carlos Sainz and Max Verstappen hustled their Toro Rossos to 11th and 12th places ahead of Felipe Nasr, who had to play catch-up after being forced to make a pit stop at the end of the opening lap due to damage on his Sauber.
The Brazilian was not alone: Lotus duo Romain Grosjean and Pastor Maldonado were also caught up in the turn one melee, with both retiring on the opening lap as a result. Hulkenberg, also involved in the chain of minor collisions, complained of damage at the front of his car throughout the race.
Over at McLaren, Jenson Button was coming under increasing pressure from Fernando Alonso, only for the Spaniard to retire after 47 laps. The Briton went on to finish 14th, ahead of Marussia's Will Stevens and Roberto Merhi.
As it result Hamilton moves onto 252 points, with Rosberg on 199 and Vettel on 178. Mercedes, meanwhile, sit on 451 points, to 270 for Ferrari and 188 for Williams.