The former Chelsea man, who missed a penalty against his old club in the final last year, found the net with just 90 seconds remaining to settle an end-to-end contest packed with drama and incident to secure a fifth European crown for the Bundesliga champions.
Bayern went ahead in the 60th minute through Mario Mandzukic after being set up by Robben, but Jurgen Klopp's men responded soon after thanks to Ilkay Gundogan's penalty, while goalkeepers Roman Weidenfeller and Manuel Neuer were in superb form all night.
After a tight opening spell, Dortmund were the first to settle with Robert Lewandowski forcing Neuer to tip a venomous long-range strike over the bar and the keeper called into action again soon after to deflect Jakub Blaszczykowski's low near-post strike behind.
Bayern's goalkeeper was worked again by Marco Reus in the 19th minute when the talented midfielder evaded the attention of Philipp Lahm and Dante to warm Neuer's palms with a drive from outside the box, before Sven Bender's low curled effort was gathered with relative ease.
Bayern, who tasted final defeat in 2012 and 2010, finally threatened in the 27th minute as Mandzukic's towering header from Franck Ribery's cross was acrobatically turned over by Weidenfeller, and Javi Martinez then escaped from his marker to glance the resulting corner narrowly over.
Weidenfeller then had to race from his line to deny Robben as the Holland international burst in behind the Dortmund defence following Thomas Muller's incisive pass, before counterpart Neuer again demonstrated his ability to deny Lewandowski after he got in behind Jerome Boateng.
Muller was left to rue a missed opportunity from a corner when he failed to make a clean contact, before the half drew to a close with an unorthodox block from Weidenfeller as he kept out Robben's shot with his face after Mats Hummels misjudged a long ball over the top by Dante.
The deadlock was finally broken on the hour mark as Ribery threaded a pass through the advancing Dortmund defence for Robben to run onto, with the former Chelsea man drawing Weidenfeller to the touchline before cutting the ball back across the box for Mandzukic to tap home.
But Bayern's lead was short-lived with referee Nicola Rizzoli pointing to the spot after Dante's clumsy lunge on Reus, allowing Gundogan to step up to send Neuer the wrong way in front of the yellow wall of Dortmund fans.
Jupp Heynckes' side looked certain to regain the lead only to be denied by a stunning last-ditch clearance from Neven Subotic, who appeared second best to Robben but showed the commitment and desire required to clear Muller's pass from off the toes of the Bayern man.
Lewandowski rifled a dipping volley home from long range but Rizzoli's whistle had already been blown for handball against the striker, before David Alaba's drive towards the top corner was fisted away by a diving Weidenfeller, with the Dortmund keeper in action once again as the 90 minutes ran down to deny Schweinsteiger.
But just when the game looked certain to go into extra time, Robben burst through the centre of the Dortmund defence to collect Ribery's back-heel, jinked away from one tackle and Weidenfeller before rolling the ball home.
After the match, Bayern Munich hero Robben admitted winning the UEFA Champions League buried his previous demons.
He told Sky Sports: "There are so many emotions, especially where we came from last year and such a disappointment. The last four years, Bayern have been in the final three times. It needed to happen but you still have to do it.
"It is a special feeling. You cannot describe it. Everything is going through your mind. It is hard to say but you do not want to be a loser."
And Dortmund defender Hummels admitted Bayern deserved to be crowned champions of Europe after their dramatic victory at Wembley.
He told Sky Sports: ""The game was really close, but it was kind of deserved for Bayern because they were the better team in the second half, but for us it is very hard and disappointing.
"They get the cup, we don't. It is one of the hardest games we can lose."