In between, Neuer produced a superb last-ditch block to thwart Lewandowski, who had deftly rolled Jerome Boateng from a Reus pass, as the play passed from one end to the other at breathless speed.
It looked as if another major final was set to pass Robben by, but on the hour he atoned for his earlier misses by creating the opening goal.
Ribery rolled a pass toward the byline and Robben evaded the offside trap, before nudging the ball beyond Weidenfeller and crossing for Mandzukic to hook home from close range.
Dortmund might have been forgiven for feeling deflated given their first-half exertions, but instead they drew level.
Dante was penalized for an untidy high foul on Reus inside the area and Gundogan steered his penalty into the bottom-right corner to send the hordes of yellow-shirted fans behind Neuer’s goal leaping from their seats.
Moments later, Dortmund were indebted to a breathtaking piece of defending by Neven Subotic, who slid in to hook Mueller’s goal-bound shot off the line as Robben closed in for a tap-in.
However, Bayern continued to push, Weidenfeller repelling fierce strikes from David Alaba and Bastian Schweinsteiger, before Robben tiptoed through the Dortmund defense to score a famous late winner.