The German ace went wheel-to-wheel with Webber late in the race, earning a stern rebuke from team principal Christian Horner, before emerging in front and taking his 27th race win.
Former world title-holder Lewis Hamilton was third, claiming his first podium spot for new team Mercedes, after his frustrated stablemate Nico Rosberg was ordered not to overtake as they duelled in the closing stages.
Ferrari's Felipe Massa finished fifth, stealing the limelight from Fernando Alonso after the two-time world champion went out on the second lap with a collapsed front wing.
France's Romain Grosjean was sixth, outshining his Lotus team-mate and last week's winner Kimi Raikkonen in seventh, with Nico Hulkenberg, Sergio Perez and Jean-Eric Vergne rounding out the top 10. McLaren's Jenson Button was a late retirement.
Webber and Vettel exchanged words in the team garage and the Australian was stony-faced during the victory ceremony, spraying little of the customary champagne.
Later, Vettel offered a profuse apology to his team-mate. Both drivers had been told by their team to hold their positions until the end of the race.
"I'm not entirely happy -- I think I did a big mistake," Vettel said. "We should have stayed in the positions that we were in. I didn't ignore it on purpose but I messed up in that situation and obviously took the lead from Mark.
"I can see now that he's upset. I want to be honest at least and stick to the truth. I know that doesn't really help his feelings right now."
A heavy tropical shower in the hour before the race greased up the track to such an extent that several cars including Webber's slid off in the pre-formation drive.
Massa dropped back off second place and Alonso damaged his front wing on the Red Bull of pole-sitter Vettel at an otherwise ultra-careful start.
But there was disaster for Ferrari's Alonso when the wing folded under his front wheels as he duelled with Webber on lap two, leaving him sliding helplessly out.
Webber moved into the lead when Vettel pitted for medium tyres on lap six, quickly followed by Massa, and the Australian remained there after his team told him "stay out, it's too wet". He finally switched tyres on lap nine.
Hamilton provided a moment of hilarity when he briefly pitted in the garage of his ex-team McLaren, before being waved on to the Mercedes crew by his bemused former colleagues.
Caterham's Charles Pic lost his front wing in a pitlane collision with Toro Rosso driver Jean-Eric Vergne, and Pastor Maldonado ploughed into a gravel trap on lap 14.
Webber regained a short-margin lead from Vettel after their second pitstops and the two Red Bulls battled with each other, with Hamilton close on the current world champion's tail, as the 56-lap race reached halfway.
Hamilton nearly clipped Vettel as he came in for his third stop and as he was soon followed by the other leaders, Button was briefly first -- before a botched pit stop, when an unsecured front wheel effectively ended his challenge.
Webber was now back in front and Vettel used his DRS to fire past Hamilton and re-form the Red Bull one-two on lap 39, ahead of the Briton and his Mercedes team-mate Rosberg.
Webber then pitted after Vettel for his fourth and final stop and he managed to regain just in front of his team-mate as they raced neck and neck on lap 44.
Vettel threw caution to the wind as he tried to take Webber down the home straight, driving perilously close to the barriers and finally going in front after some hair-raising action around several turns.
"This is silly Sebastien," barked team boss Christian Horner over the team radio, as Vettel led with 10 laps remaining.
Hamilton was grittily defending third from Rosberg, who complained he could go faster than the 2008 world champion -- but was also told to rein it in by team principal Ross Brawn. "Please look after the cars," Brawn pleaded.
Afterwards, Hamilton admitted he didn't feel "spectacular" to be on the podium rather than his team-mate Rosberg. "It isn't the best feeling to be up here," he said during the victory ceremony.
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