In a game that featured a slew of chances for both teams, Mario Mandzukic put Bayern ahead in the 60th minute at Wembley Stadium before Ilkay Gundogan evened the score on a penalty kick eight minutes later after defender Dante fouled Marco Reus in the area.
Robben, who missed two great chances in the first half, then finally came through for Bayern to set off wild celebrations in the red-and-white end of Wembley. It was the perfect redemption for Robben, who missed a penalty in extra time of last year's final, causing some Bayern fans to turn against him. A year later, after the final whistle sounded in London, he strode with his arms raised toward the section of jubilant Bayern fans.
Robben also set up the first goal for Bayern, taking a pass from Ribery and drawing Weidenfeller out toward the sideline before squaring for Mandzukic, who could hardly miss from a few yards out.
But the lead didn't last long. Dante clumsily clattered into Reus in the penalty area, and Italian referee Nicola Rizzoli pointed to the spot. Gundogan sent Manuel Neuer the wrong way before slotting his kick into the right side of the net.
Bayern entered the game as the clear favorite, although that message didn't seem to have filtered through to Juergen Klopp's Dortmund team, which seemed intent on dictating play from the start. It largely succeeded for the first 25 minutes as well, although it was Bayern that ended up with the best chances of the first half.
Mandezukic came closest when his header was tipped onto the bar by Weidenfeller in the 26th, and the goalkeeper then denied Robben four minutes later when the speedy winger raced clear toward the goal.
Before that, Neuer had been by far the busier goalkeeper, making a point-blank save from Jakub Blaszczykowski and diving to stop an effort from Sven Bender. Neuer also pushed away long efforts from Reus and Robert Lewandowski before denying the Polish striker again from close range in the 35th.
Robben had the final chance of the half when he beat Mats Hummels to a long ball into the area in the 42nd, but fired his volley straight into Weidenfeller's face.