The Red Bull driver started second on the grid and was beaten to the first corner by Hamilton, but Vettel used his KERS system to blast past his rival's Mercedes going into the Les Combes chicane, just the fifth corner of the race.
Vettel never looked back, and there were no major on-track incidents or rain - despite grey skies and a gloomy forecast - to derail his procession to a fifth win of the season. The result stretches his championship lead to 46 points.
"It was a fantastic race for us from start to finish really," he said after his 31st career win.
Red Bull principal Christian Horner was overjoyed with the outcome.
"He (Vettel) had the momentum and was able to go around Lewis and that was the decisive moment," he said.
"There's still a long way to go but it was a dominant performance from Seb, perfect display by the team with pit stops and strategy."
Fernando Alonso had qualified appallingly in ninth, but a superb start and great driving thereafter saw him climb to second place on the podium, while pole sitter Hamilton was a distant third in a race that turned into a tedious procession.
"Today it was a little bit boring, we get second place but no threat to Sebastian Vettel and had no threat from behind," said Alonso, whose team's home Italian Grand Prix is next up.
"It is an important weekend for us and the team and we arrived fully motivated and in Monza we would like to give some smiles to our fans."
Kimi Raikkonen suffered front brake failure, failing to finish a grand prix for the first time in 38 races, which means Alonso leapfrogs him to second place in the title race. But with Vettel almost two race wins clear of the chasing pack there now seems nothing that can stop the German winning a fourth consecutive drivers' championship.
The high point of excitement actually came before the start: a group of protestors from Greenpeace parachuted onto the roof of the main grandstand in an attempt to complain about Shell's drilling in the Arctic by unfurling a banner.
A later first pitstop brought Alonso right up behind Hamilton, who he then passed as the Mercedes slipped a touch wide at La Source.
Hamilton retaliated with DRS on the Kemmel Straight, yet Alonso was able to fend him off despite a vicious twitch under braking.
Jenson Button looked like he might have a say in the podium fight as he ran long and hinted at a one-stop strategy.
In the end he had to follow the two-stop trend, dropping him behind Hamilton, fourth-placed Rosberg and the slow-starting Mark Webber.
The Australian's Red Bull showed great late pace having used hard tyres in the middle stint and softs at the end, the opposite strategy to most rivals, but ran out of steam when he came up behind the Mercedes.
Raikkonen looked set to finish adrift of this group even before he was forced to retire his Lotus.
Felipe Massa resisted Romain Grosjean, the only successful one-stopper in the points, for seventh place.
Grosjean had an early brush with Sergio Perez in which the stewards judged that the Lotus had been forced off the road at Les Combes.
That earned Perez a drive-through penalty - and that plus late tyre wear on a one-stop left him 11th at the flag.
Force India's Scottish driver Paul Di Resta endured the most miserable end to his weekend: on Saturday afternoon it looked for a while as if he had stolen pole with a brilliantly-timed lap.
His luck ran out, however: after ending up fifth in the final moments of qualifying, he also ended up being taken out by Pastor Maldonado, who bounced off Adrian Sutil and into the path of Di Resta to completely detach one of his rear wheels.
Force India still scored thanks to Sutil's ninth place.
Daniel Ricciardo overcame Toro Rosso's qualifying miscue to come from 17th to 10th.