Mercedes' Lewis Hamilton dropped to last with tyre failure, but fought to fourth ahead of Lotus's Kimi Raikkonen.
Alonso's third-place finish, combined with Vettel's retirement with a gearbox failure caused by the loss of fifth gear, cut the German's advantage
in the championship to 21 points.
Formula 1: Lewis Hamilton wants action on 'unacceptable' tyre failures
Raikkonen remains third but is now 11 points adrift of the Spaniard.
The race boiled down to a seven-lap sprint at the end following a safety car intervention to recover Vettel's Red Bull, which was stranded on the pit straight.
Rosberg was in the lead from Raikkonen and Force India's Adrian Sutil, with Webber fifth, Alonso eighth and Hamilton ninth.
Rosberg, Webber, Alonso and Hamilton were all on fresh tyres, the others around them not.
Webber passed Toro Rosso's Daniel Ricciardo for fourth and then Sutil and Raikkonen on consecutive laps, and although he closed in on Rosberg the German held him off to win by 0.7 seconds.
Meanwhile, Alonso was scything through from eighth - to which he had dropped from fifth because his tyre stop had come just before the caution period, meaning he lost more time than the drivers who stopped only after the safety car was deployed. Hamilton, on his tail, matched his progress.
"Fantastic, a very special day. What makes it more special is that our factory is so close," said Rosberg. "We're progressing all the time. We're massively quick in qualifying, and we had the fastest car in this race.
"With Lewis, I feel sorry for all the British fans. It's always a massive disappointment but that's racing. Sebastian stopped, I won't lie, I wasn't disappointed by that one."
But the thrill of the battle was overshadowed by a worrying sequence of tyres failures, all but two on the left-rear of the cars involved.
They affected Hamilton, Ferrari's Felipe Massa, Toro Rosso's Jean-Eric Vergne and McLaren's Sergio Perez, who had also had a similar failure in final practice.
Gearbox woes end Vettel's British GP
There was a fifth failure, of the left-front tyre, on Esteban Gutierrez's Sauber. Alonso also suffered a failure of his right rear tyre in Stowe corner - the last corner before the pit entry - just before his first pit stop.
McLaren managing director Jonathan Neale described the situation as "very concerning", adding: "We have to ensure racing is safe. We can't afford instantaneous failures like this."
Hamilton had earlier led away from pole,
with Vettel passing Rosberg for second, only to suffer the first of the tyre failures on lap eight, going down the Wellington Straight. Massa's tyre failed two laps later, at Turn Four, a couple of hundred metres before Hamilton's.
That put Vettel into the lead, from which he controlled the race ahead of Rosberg, through and beyond the first safety car period triggered by the need to clear the debris left by Vergne's tyre failure.
But he lost drive heading into Club corner with 11 laps to go, closing up the title race.
Vettel and Rosberg were among a number of drivers whose teams discovered were also on the verge of failure when they took them off at a pit stop.
"It definitely needs to be looked into," Rosberg said, "because there were too many today."
Behind Vettel and Rosberg, the race was characterised by Raikkonen and Alonso making ground from their starting positions of eighth and ninth.
Webber recovered from a poor start that dropped him to 13th on the first lap - on which he also suffered a damaged front wing after being hit by Lotus's Romain Grosjean at the start - and Hamilton fought back from his tyre failure.
"I didn't have a clue what happened off the line," said Webber. "The lights went out and we were back to usual tactics. That was frustrating.
"I think it was Grosjean who took my front wing off in the first corner. Boys did a good job changing the front wing at the first stop.
"It's my last time here for the British fans, so thanks."
It was a grand prix full of close racing and incident but the over-riding impression will be of tyres that were not up to the task.
Pirelli had introduced a new bonding process for this race in an attempt to prevent a series of delaminations that happened through the season.
There have now been 20 tyre failures so far this season.
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