
Having been crowned German league champions on Sunday, Bayern missed all four of their spot-kicks to put Dortmund in the German Cup final on May 30.
Bayern — who did win the treble of European, Cup and Bundesliga titles in 2013 — can still win the Champions League as they face Barcelona next Wednesday in the semi-final first leg.
"The shoot-out had a stupid start, it summed up our game," said goalkeeper Manuel Neuer, who failed to convert Bayern's fourth penalty. "We could have decided it over 90 minutes and also in extra-time. It's a bitter defeat."
Robert Lewandowski scored against his former employers to give Bayern a first-half lead before Dortmund striker Pierre-Emerick Aubameyang equalised 15 minutes from time to take the game into extra-time.
The shoot-out got off to a dramatic start when Philipp Lahm and Xabi Alonso both slipped to fire Bayern's first two penalties over the bar.
Ilkay Gundogan and Sebastian Kehl converted theirs for Dortmund before Mario Goetze had his effort saved by Borussia's Australia goalkeeper Mitchell Langerak.
Dortmund captain Mats Hummels had his shot saved before Munich's shot-stopper Manuel Neuer hit the cross bar with Bayern's fourth spot-kick to send Borussia to Berlin.
"That was a hell of a game, both teams gave everything and Bayern were certainly superior in the first-half, but we more than deserved to equalise in the second-half, today was our day," said Dortmund coach Jurgen Klopp.
Fatigue cost Bayern in shoot-out, admits Pep
Meanwhile, Bayern coach Pep Guardiola admitted fatigue ended Bayern Munich's hopes of winning the treble.
Guardiola however insisted he is as proud of his team "as never before", but admitted their punishing current schedule of a game every three or four days had taken a heavy toll.
"That was our best game against Dortmund in the period I have been in Germany, but we were a bit tired in the penalty shoot-out," said Guardiola. "My team gave it everything and I must congratulate Dortmund, I wish them well for the final."
This was the first time Bayern have lost on penalties since the 2012 Champions League final defeat at home to Chelsea.
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