The Australian finished 1.36 seconds ahead of championship leader Lewis Hamilton of McLaren, turning in an immaculate drive at Silverstone after being angered by having to hand over a key car part to Vettel.
"Not bad for a No. 2 driver," Webber quipped over his radio during his victory lap.
Nico Rosberg of Mercedes was third.
With nine races to go, Hamilton leads the overall standings with 145 points, 12 ahead of McLaren teammate Jenson Button. Webber is third with 128 after becoming the first driver this season to win three races. Vettel is fourth with 121.
Webber's car has been equipped with the old version of Red Bull's front wing since qualifying Saturday after the new design was stripped and handed to Vettel, who subsequently took the pole.
But Webber led throughout Sunday after pushing Vettel wide on the opening lap, when Vettel also punctured a tire.
"I was obviously keen to make it my corner and it worked out well for me," Webber said. "The car was faultless all day."
But amid the celebrations, Webber made clear his unease at the current situation.
"Yesterday, I wasn't happy clearly," Webber said. "I would never have signed a contract for next year if I believed this would be the way going forward."
Vettel, who won the British GP last year, had been forced to the back of the field after the early setback. After the safety car emerged on lap 31, Vettel was able to pick up pace and start overhauling his slower rivals and finish seventh.
"I had a bad start with lots of wheelspin, so I lost the immediate pull away, probably trying a bit too hard, but that's life," Vettel said. "I went into turn one and had to let Mark past. People said that Lewis [Hamilton] touched me but I didn't feel anything.
"Mark had a better start. I moved to the right trying to defend, but he was already there. There was no point to do something stupid, the race is longer than just one corner. It's a shame I couldn't fight in there, but I am happy for the team."
The main beneficiary of Vettel's disastrous start was Hamilton, with the 2008 champion moving up two places to second ahead.
Rosberg's third place was the German's best showing since China in April and deprived Button of his first podium finish on home soil. Button, the defending world champion, climbed 10 places from 14th on the grid to finish fourth.
Button had branded his car undriveable after flawed upgrades were abandoned on Friday night. The Englishman was helped in the race by Robert Kubica after the Renault driver, who started from sixth, retired for the first time this season with a rear axle problem after 19 laps.
"It's so difficult overtaking out there so I knew I had to make a big amount of places up on the first lap -- I made six places up -- and then we made the right call with pitting late," Button said.
Fernando Alonso of Ferrari finished 14th after the double-blow of a drive-through penalty for going onto to the grass to overtake Kubica and a late puncture after touching Force India's Vitantonio Liuzzi
The other drivers to score points were Rubens Barrichello of Williams, who was fifth, followed by Sauber's Kamui Kobayashi, who was ahead of Vettel, Adrian Sutil, Michael Schumacher and Nico Hulkenberg.
Schumacher's two points were his first since finishing fourth in Turkey in May.