With Vettel's Red Bull teammate Mark Webber closing fast, Rosberg managed to win his second race of the year and third of his career by 0.7 seconds in a nail-biting finish - though he then had to survive a stewards inquiry.
Ferrari's Fernando Alonso, who started ninth, made a late charge up the grid and got past Lotus driver Kimi Raikkonen with two laps remaining to claim third. Hamilton also recovered to finish fourth, after his blown tire sent him to the back of the grid in the eighth lap.
Soon after celebrating, Rosberg risked seeing his victory slip away when he was referred to the stewards for not slowing down for yellow flags in turns 3 and 5. The stewards ruled that Rosberg "did not make a significant reduction in speed" but would only receive a reprimand, or warning. After three reprimands during a season, drivers are given a 10-place grid penalty. This was Rosberg's first.
"Fantastic, it's very special," said Rosberg, who also won in Monaco and in China last year. "Our team has done such a fantastic job. We have such momentum at the moment, progressing all the time."
Hamilton seemed poised to win his first British Grand Prix since 2008 after getting a great start and extending his lead over three-time world champion Vettel.
But on the eighth lap, the 2008 champion's left rear tire exploded and he was forced to limp into the pits. Two laps later, Ferrari's Felipe Massa lost his left rear tire and spun out. Then, Toro Rosso's Jean-Eric Vergne also lost his rear tire on the 15th, with the rubber exploding across the track.
That brought out the safety car until lap 22 and sparked renewed concerns about the reliability of Pirelli tires which has been a running theme all season. It also prompted drivers to later warn of the dangers of exploding tires - which affected McLaren driver Sergio Perez on both Saturday and Sunday.
Alonso admitted he was "scared" after being sprayed with debris from Perez's tire and Rosberg called for action to prevent a similar situation in Germany next weekend.
"Well, we shouldn't get into that situation," Rosberg said. "We need to do what needs to be done to sort it out and make the tires last."
The FIA has ordered tire supplier Pirelli to attend a meeting on the issue on Wednesday in Germany.
Pirelli said it was investigating the matter, but ruled out a new bonding process for the tires that was introduced ahead of the British Grand Prix being to blame. The company said "there may be some aspect of the circuit" that contributed to the problem.
"There have obviously been some issues with rear-left tire failures which we have not seen before," Pirelli's Motorsport Director Paul Hembery said. "We are taking the situation very seriously and we are currently investigating all tires to determine the cause as soon as possible, ahead of the next Grand Prix in Germany."
Hamilton was still clearly angry after the race.
"The safety is the biggest issue, it's unacceptable," Hamilton told the BBC. "Four blowouts is unacceptable. It's only when someone gets hurt that someone will be doing something about it. I think it's a waste of time talking to the FIA, and if they don't do anything that says a lot about them."
TV pictures of Turn 4 on the Silverstone circuit, where some of the incidents took place, indicated the sidewall of the tires could have been punctured by a sharp part of the kerb.
After the restart, Vettel extended his lead over Rosberg and appeared set for a fourth win which would have extended his lead in the drivers' championship. But on the 42nd lap, Vettel's car came to a stop after his fifth gear broke and Rosberg took the lead. That brought out the safety car for a second time for three laps.
When it went off with seven laps to go, it triggered a frenzied dash to the finish.
Alonso was among the most aggressive, passing McLaren's Jenson Button, Force India's Adrian Sutil and eventually Raikkonen for third. Webber also charged past Raikkonen into second with four laps remaining. Raikkonen later said his tires were too worn for him to fend off the challenges of Webber and Alonso.
Webber — who announced Thursday he was retiring from the sport at the end of the year — just missed out on his third win at Silverstone in four years and a fairytale ending to the weekend.
"I made the most of those safety cars — and today the strategy was one of the best we've done," said Webber, who dropped down to 15th at the start after getting off the grid slowly and then being nudged off the track by Lotus driver Romain Grosjean.
"I think it worked very, very well. I worked hard to manipulate the pace when I had to and then it was Nico and I left at the end," he said. "It would have been nice to have had a few more laps, but that's how it was today. It was nearly a really special day, but it was still a nice day."
Red Bull team principal Christian Horner said he was happy to come away with some points, considering the kind of race his team had.
"Obviously to get within 12 laps of winning the race and have a retirement due to a transmission issue is pretty painful, particularly at this venue, but Sebastian had done everything right today and it was a great shame," he said. "For Mark, after a difficult start and being hit at the first corner, he made a great recovery ... and came pretty close to claiming the victory, but nonetheless second place was a great performance from him on a big weekend for the team."
Hamilton also benefited from the safety car going out twice, rising from last on the grid to ninth by lap 45 and then got past Sutil for fourth with a lap remaining.
"There are massive positives to take out the team this weekend," Hamilton said. "Nico won and we got some good points and we are second in the constructors' championship. I'm grateful the car was strong I was able to come through the field and the right things happened. It was bad luck when the tire goes and you are in the lead. I was grateful that I had achieved things that helped me along the way."
The wild finish leaves the championship wide open.
Despite his late exit, Vettel remains the leader in the drivers' championship with 132 points. His lead, though, is down to 21 points over Alonso on 111 with 11 races remaining.
Raikkonen is third with 98 and Hamilton fourth with 89, some 43 points behind Vettel.
The next race is the German Grand Prix on July 7.