The hosts struck three times in the opening 21 minutes as they demolished the German champions and record 20-time Cup winners.
It was Bayern's biggest defeat in a domestic game in 43 years since a 7-1 Bundesliga meltdown at Fortuna Dusseldorf in December 1978.
Gladbach's dream start yielded two goals from Ramy Bensebaini and one from Kouadio Kone to put them in the driving seat before Breel Embolo struck twice early in the second half to kill off the game.
They became the first team to score five goals against the mighty Bavarians since Borussia Dortmund's 5-2 win in the 2012 German Cup final, and the first ever team to score three goals in the first 21 minutes against Bayern in a German Cup game.
"It is shocking. It was a collective blackout. I have no explanation for this," Bayern board member sport Hasan Salihamidzic told broadcasters ARD after the humiliation.
Gladbach ousted Bayern for the first time in the competition in the teams' seventh meeting.
"The night will surely become part of Borussia Monchengladbach's history," Gladbach sports director Max Eberl said.
Bayer Leverkusen were another Bundesliga side to bow out,, crashing 2-1 at home against second-division Karlsruhe.
Elsewhere, VfL Bochum beat Augsburg 5-4 on penalties after a 2-2 scoreline over 120 minutes.
Anthony Modeste scored twice in five minutes after coming on in Cologne's 2-0 at VfB Stuttgart while Union Berlin came from behind to win 3-1 in extra-time at third-division Waldhof Mannheim.
Second division leaders St Pauli were among others to adnace.
with DPA
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