Following a goalless 120 minutes, Bayern goalkeeper Manuel Neuer, who was in fine form throughout the game, saved a penalty by Havard Nordtveit after Gladbach's Dante had fired his spot-kick over the bar to clinch a 4-2 win on penalties.
Bayern are also still in the running for the Bundesliga and Champions League titles.
"Both teams played an outstanding game," Bayern coach Jupp Heynckes told reporters. "We had more chances but it was an open game until the end. "Manuel Neuer has to be singled out for a superb performance."
They were lucky, however, not to concede a goal after six minutes with Toni Kroos' low drive hitting the post.
Bundesliga top scorer Mario Gomez was then denied by Gladbach keeper Marc Andre ter Stegen, who parried the forward's powerful header.
In an entertaining encounter, Bayern, who had scored 20 goals in their last three games, had the better chances with Arjen Robben twice coming close while Gladbach's Marco Reus was also twice denied by Neuer, including six minutes from time in a one-on-one situation.
"It is tough being eliminated on penalties," said Gladbach coach Lucien Favre, whose team are third in the Bundesliga. "But we have to look forward. We have to get up and because Saturday is again an important game for us. We can't change what has happened."
