Frenchman Ribery's backheel allowed Dutchman Robben to break through a gap in the Dortmund defence and dance round goalkeeper Roman Weidenfeller to score the 89th minute winner.
The goal sent Bayern's fans into raptures as their team became European champions for the fifth time to match Liverpool's haul of titles.
Only Real Madrid with nine triumphs and AC Milan with seven have been crowned champions more often.
Bayern soaked up almost non-stop Dortmund pressure before going ahead in the 60th minute when Ribery and Robben again combined in the build-up to set up Croatian Mario Mandzukic to hook in with his left foot from close range from a Robben cross.
Dortmund, who fashioned seven scoring chances in the first half, pulled level when Bayern defender Dante clumsily kicked Marco Reus in the stomach eight minutes later with Ilkay Guendogan sending Manuel Neuer the wrong way with his penalty.
But just as the match appeared destined for extra time, Robben, who missed a penalty in last season's final, pounced to collect Ribery's clever backheel and score the winner.
"So many people told me I'd score a goal tonight. the first few chances I missed. I got another chance at the end and took advantage of it," said Robben.
"It means a lot to me. It's still hard to fathom. I'll be able to put it in words later. There are so many emotions right now, it's incredible."
The victory eased the pain of Bayern's defeats by Inter Milan in the 2010 final and the heartbreaking defeat to Chelsea on penalties in their own stadium a year ago as they lifted the European Cup for the first time since 2001.
Bayern's success puts them in touching distance of an unprecedented treble for a German club as they have already been crowned Bundesliga champions and play VfB Stuttgart in the domestic cup final in Berlin next week.
It also means Bayern's 68-year-old coach Heynckes, who is leaving at the end of the season, becomes only the fourth manager to win the European Cup with two different clubs following his earlier success with Real Madrid in 1998.
He joins an elite group of Ernst Happel, Ottmar Hitzfeld and Jose Mourinho and either his tactics or fortune stood him in good stead at Wembley where Bayern finished the stronger side after Dortmund threatened to overrun their opponents in a blistering first half exhibition of attacking football.
For their part, Bayern, coming back off the ropes, attacked almost as much and between them the two sides produced 12 scoring efforts in the opening 45 minutes with Bayern creating five good chances - three of them falling to Robben.
However, both goalkeepers were in superb form with Weidenfeller saving everything Bayern conjured up while Neuer was just as eye-catching at the other end in the first all-German European Cup final.
Dortmund, looking to add a second European crown after their 1997 victory, played Bayern at their own game with a high intensity, penetrative attacking plan devised by their impressive young coach Juergen Klopp.
Next season he will have to pit his wits against Pep Guardiola who will inherit a Champions League-winning side who just had a little too much depth and experience for the losers.
"It was a dramatic match. A close match," said Klopp. "It was a long season and we were all exhausted. Congratulations to Bayern, and to Jupp. We wanted to win the thing today... but that's the way it goes and we can't change it."
Dortmund began at blistering pace which they maintained for almost 30 minutes with Reus and their Poland striker Robert Lewandowski pulverising the Bayern defence.
Lewandowski had the first of their seven scoring chances in the opening period, powering in a 30-metre shot which Neuer turned over the bar for a corner.
Neuer also made a superb stop when he stuck out his left leg to prevent Jakub Blaszczykowski scoring from close range after he was set up by Lewandowski.
Bayern were virtually over-run at the start but battled back to create five good scoring chances of their own - three falling to Robben who could well have netted a hat-trick by halftime.
A poor touch scuppered the first when he had only Weidenfeller to beat, and the goalkeeper stopped him again by taking a shot straight in the face when Robben had the goal gaping just before the break.
Weidenfeller also stopped a powerful goalbound header from Mandzukic after 26 minutes, tipping his powerful effort on to the bar and over.
The only surprise was that neither side made the breakthrough after 45 breathtaking minutes of superb football.
The game continued to be end-to-end for most of the second half, but Bayern gradually took control as Dortmund faded with Lewandowski failing to have the kind of impact he made in the semi-final when he scored four times against Real Madrid.
Reus also faded as Bayern's defence tightened up, while Ribery and Robben kept powering forward and in the end their pace and guile came together to win the day, leaving the players to celebrate with the big trophy and the departing Heynckes.