The Dutchman touched in from Jerome Boateng's cross in the second minute of the second half of extra-time after a tight match in Berlin.
Thomas Muller broke clear in added time to score a second to seal the double in Pep Guardiola's first season in charge.
Earlier a header from Dortmund's Mats Hummels from a Nuri Sahin free-kick appeared to cross the line.
Bayern finished 19 points above second-placed Dortmund in the Bundesliga but there was little between the sides until Guardiola's team showed ruthlessness in the final quarter of an hour to continue their domestic dominance.
They had the best chance of the first half after just two minutes when Robben threaded a pass through for Muller, whose powerful effort bounced away off goalkeeper Roman Weidenfeller's face.
Robert Lewandowski, who will join Bayern from Dortmund next month, sliced high and wide with his only opportunity just before the break.
Hummels appeared to have the ball over the line before Dante hacked away in the second half.
Bayern finally broke through when Kevin Grosskreutz failed to control Weidenefeller's throw and Boateng crossed for Robben to stab in.
Robben hit the post with a low drive before Moller broke away and ran around Weidenfeller before rolling into an empty net in added time at the end.
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