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.
It was also their first ever Cup win over Bayern.
“I am absolutely shocked,” said Bayern sports director Hasan Salihamidzic. “We were just not there. We did not win a single tackle or challenge in the entire first half.
“Everything that could go wrong went wrong. Gladbach did it really well. We were absolutely not there. A blackout. It should not have happened to us in such a way.”
The Bavarians, who have scored a record 33 goals in nine league games and have added another 12 goals without conceding any in three Champions League matches, were the overwhelming favorites.
With the Bayern defense in complete disarray, the hosts should have scored again with Jonas Hofmann and Embolo coming close as the defensive efforts by the visitors piled up.
Gladbach did it better in the 15th, outplaying the static Bayern backline for Bensebaini to thunder a Hofmann cutback past Neuer.
Before they had any time to recover, Lucas Hernández, who earlier on Wednesday had seen a six-month prison sentence suspended in Spain, clumsily brought down Embolo in the box and Bensebaini sent Neuer the wrong way with an audacious spot kick to make it 3-0.
With Bayern coach Julian Nagelsmann confined to his home after testing positive for Covid-19, assistant coach Dino Topmoeller had to wait until the second half to see some decent attacks from his players.
But before they even got into any rhythm, Gladbach struck again with man-of-the-match Embolo beating two defenders, who let a high ball bounce between them, to slot the ball in through the legs of Neuer.
The Swiss bagged his second goal of the evening in the 57th, finding far too much space once more to score their fifth off the post.