Jupp Heynckes' men were given a run for their money by the Foals, but ultimately did just enough to edge past the home side and book a place in crowning game of the tournament.
After two action packed periods and half an hour of extra time, the game went to the lottery of spot kicks.
Bayern Munich proved to have nerves of steel, netting all its penalties, while Havard Nordtveit and Dante missed for Gladbach as the club's fairytale in the DFB Pokal came to a timely end.
Bayern Munich started the game aggressively, creating the first chance within just six minutes after the first whistle. Toni Kroos latch onto a clearance by Dante to smash a superb shot into the left post from just outside the box.
The Bavarians did very well to absorb any pressure Lucien Favre's men were trying to apply, but could not really find enough space to pierce the Foals' watertight defense. Arjen Robben and Franck Ribery looked isolated on the flanks, and were not seeing much of the ball.
Borussia Monchengladbach created its first real chance of the game in the 20th minute when a precise through ball by Roman Neustadter released Marco Reus on the left side of Bayern Munich's defense, but the Germany international was denied superbly by Manuel Neuer.
Jupp Heynckes' men, however, slowly regained the upper hand and dominated the game. In the 40th minute, Ribery exploded into life, leaving Tony Jantschke reeling on the wing before finding Robben, who could only fire over the bar from close range.
Moments later Bayern Munich's left flank once again bypassed Monchengladbach's defense, with David Alaba sending a superb shot from a tight angle, only to see it pushed away by Marc-Andre ter Stegen in goal for Favre's men.
Not much changed after the restart as the two teams looked evenly match. However, it was Bayern Munich which created the first chance of the second period as a lovely one-two between Thomas Muller and Robben culminated in the Dutchman once again failing to hit the target.
The Bavarians were almost punished for this miss in the 49th minute when Reus found himself in a one-on-one with Neuer, managed to hold off the goalkeeper and squared for Juan Arango, who blasted over the bar when it looked easier to score.
The tempo dropped in the following minutes as extra time seemed imminent. In the final moments of the second period, though, each team had a chance to open the scoring, but Gomez and Reus were guilty of unflattering misses.
Bayern Munich dominated the two halves of extra time, but Monchengladbach held on and the game trickled into penalties. Alaba, Riberya and Lahm netted for the Bavarians, while only Filip Daems and Patrick Herrmann converted for the Foals as Jupp Heynckes' men progressed to the final.
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