Webber led from pole to flag, but Hamilton had out-raced Sebastian Vettel in the other Red Bull before a puncture forced the Englishman out.
That promoted Fernando Alonso's Ferrari to second, ahead of Vettel, who also hit trouble in the closing stages.
Mercedes's Michael Schumacher was fourth from Jenson Button's McLaren.
Vettel looked to have lost third place when he ran into problems in the closing laps.
The German came into the pits after running off the track with 12 laps to go following a brake failure, and when he rejoined, he was warned by his team that the problem was "critical" and that he needed to nurse his car to the finish.
Felipe Massa's Ferrari took sixth, just over a second behind Button, while Force India's Adrian Sutil held off the Renault of Robert Kubica for seventh.
Rubens Barrichello was ninth for Williams after driving a superb first lap to jump up from 17th on the grid to 12th, with Toro Rosso's Jaime Alguersuari promoted to 10th by Hamilton's misfortune.
The result has tightened an already close championship battle.