The defending three-time world champion, who started from pole position, attacked the championship leader who defended hard at Turn Two and drove into him.
Rosberg forced Hamilton off the track, but the Briton rejoined, the pair touching briefly again, and pulled clear to win. Behind him, Rosberg suffered a damaged front wing and dropped back to finish fourth.
The collision concluded a titanic contest between the two Mercedes men after an incident-packed race that left Mercedes team bosses describing their drivers as 'brainless'.
Rosberg was fourth ahead of Australian Daniel Ricciardo in the second Red Bull, Briton Jenson Button of McLaren Honda, Frenchman Romain Grosjean of Haas and Spaniard Carlos Sainz of Toro Rosso.
The stewards announced an inquiry into Rosberg's part in the final lap collision for continuing with a damaged car as Hamilton was jeered by the crowd as he stood on the podium.
"I was on the outside," said Hamilton, on team radio. "It wasn't me that crashed."
It was the 46th win of his career.
"What an incredible race," said Hamilton, amid the jeers. "It was tough. Apart from this (the crowd reaction), I love it here... I don't know what that's about. It's not my problem. It's their problem."
- Near-perfect getaway -
========================
After four poor previous starts from pole this year, Hamilton made a near perfect getaway and eased clear while Hulkenberg, from second, fell back. Button grabbed second ahead of Raikkonen.
By then, Button, who had held Raikkonen behind him for six laps, was falling away and the Ferraris were closing in, Raikkonen running second and Vettel, on his 29th birthday, third by lap 16 with Rosberg, on new super-soft tyres, fourth, but 29 seconds adrift of the leader.
Tyre wear and strategy were clearly key factors in the contest.
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