Hamilton, last year's winner in Belgium, again started from pole position after a wet qualifying but his Mercedes could not match the German's pace in the dry and he had to settle for second.
Red Bull's Max Verstappen gave his army of Dutch fans something to cheer by taking third place.
In the enthralling battle between the four-times world champions, Hamilton now has 231 points to Vettel's 214 with both on five wins apiece for the campaign.
"I had a great start, I'm not sure Lewis saw me as he pushed me to the left in the first lap," said Vettel of his decisive pass on Hamilton on the Kemmel straight after the title frontrunners had made clean getaways.
"As soon as I was ahead I relaxed. I had a good restart after the safety car, and after that it was a very smooth race," he added.
MAIN DRAMA
The first corner otherwise provided the main drama, with Fernando Alonso's McLaren flying over Charles Leclerc's Sauber at the La Source hairpin after being rammed by Renault's Nico Hulkenberg.
All three retired as the safety car was deployed for four laps but Leclerc had reason to thank the 'halo' head protection system -- introduced this season -- for allowing him to step away unscathed.
Hulkenberg was later handed a 10 place grid penalty for next weekend's Italian Grand Prix.
Television close-ups after the race showed the structure heavily marked by the impact.
"We can end the HALO discussion now. It will save lives," tweeted 2016 world champion Nico Rosberg after seeing the images.
Ferrari's Kimi Raikkonen, a four-times winner in Belgium, and Red Bull's Daniel Ricciardo also collided at the start and retired after pitting, the Finn having to wait for his 100th career podium.
Vettel had an untroubled afternoon once the safety car returned to the pitlane and took the chequered flag 11.0 seconds clear of Hamilton.
"He drove past me like I wasn't even there on the straight," said Hamilton, who was quick to shake his rival's hand and offer congratulations.
"They have got a few trick things going on in the car," he added. "I did what I could, we did what we could so we have to keep working."
The Briton later clarified his remark to emphasise that he was not questioning the legality of the Ferrari.
BEST OF THE REST
The victory was Vettel's 52nd, sending him ahead of French four-times champion Alain Prost on the all-time race winners' list.
Hamilton's team mate Valtteri Bottas finished fourth, and set the fastest lap, after starting 17th.
Force India began their existence under new ownership with a haul of points that lifted them immediately off the bottom of the standings.
The team, who had started with all their constructors' points taken away following confirmation as a new mid-season entry, had Mexican Sergio Perez fifth and Frenchman Esteban Ocon sixth.
"To be 'best of the rest', with Esteban right behind me, is the most we could have done," said Perez.
"We have scored some important points for the team and sent a strong message for the next few races. It's a nice start of a new era for the team."
Haas duo Romain Grosjean and Kevin Magnussen were seventh and eighth, with France's Pierre Gasly ninth for Toro Rosso and Sweden's Marcus Ericsson 10th for Sauber.
McLaren's Belgian driver Stoffel Vandoorne was last in his home race.
(Reporting by Alan Baldwin, editing by Toby Davis and Christian Radnedge)
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