Defending three-time champion Hamilton reeled off his fourth win in five races as he steams up behind Rosberg in the title race, with the German also subject to a post-race inquiry into radio dialogue with the team when he had a gearbox problem.
Talented Dutch teenager Max Verstappen finished third -- pushing Rosberg all the way -- ahead of his Red Bull team-mate Daniel Ricciardo, with Finn Kimi Raikkonen coming home fifth for Ferrari.
"A great job and thank you so much guys for your efforts back at the factory," said Hamilton on his victory lap.
From the podium, he added: "We've got the best fans here so thank you so much... I am not sure you can be as happy as I am."
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A downpour shortly before the race created some havoc and an immediate call for "full wet" tyres and a plodding safety car start.
On lap five, he made it more obvious: "We can go, Charlie!" he said, a message directed at the race director Charlie Whiting.
And so racing began on lap six, Hamilton throwing up a high plume of spray ahead of Rosberg as most of the rest -- bar the top four -- rushed to the pits for intermediate tyres.
In the pit lane, there were two near collisions before the race resumed and settled with Hamilton pulling 3.6 seconds clear by lap seven.
As the track improved, Verstappen and then Rosberg picked up the pace before Hamilton responded. By lap 12, he was 5.2 seconds clear with Rosberg 1.4 clear of Verstappen.
Sebastian Vettel was the first in for slick tyres after 15 laps when, in an outrageous move, Verstappen swept round Rosberg at Becketts to take second.
- Radio dialogue -
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The top order stayed unchanged as the top three traded fastest laps while the circuit remained treacherous, with both Spaniards Fernando Alonso and then Carlos Sainz spinning off at Abbey, but surviving.
Hamilton almost did the same, but retained control as his car twitched. Verstappen also slithered and the top three were separated by only six seconds.
By lap 25, Hamilton was surging again, 5.9 seconds clear of Verstappen, who was defending against a rampant Rosberg.
Rosberg finally passed Verstappen on lap 38 and began to hunt down Hamilton, 8.2 seconds ahead.
The German swiftly cut that to 7.6 with 11 laps remaining while compatriot Vettel, who had forced Felipe Massa off a few laps earlier, was handed a five-second penalty.
At the front, Rosberg reeled off fastest laps, but Hamilton nursed his car, and the tyres, as he retained a six seconds' lead until, on lap 47, Rosberg reported a gearbox problem. He lost three seconds and fell back to 11 seconds behind Hamilton.
Sensing blood, Verstappen closed up, notably when the stewards launched an investigation into Rosberg's radio dialogue, but in the end Rosberg hung on.