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 was also the first time Bayern Munich lost a game by five or more goals since Dec. 9, 1978 against Fortuna Düsseldorf (7-1) in the Bundesliga.
"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 favourites.
With the Bayern defence 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 Hernandez, 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, who have netted a record 33 times in their first nine league games.
But before they even got into any rhythm, Gladbach struck again with 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.
The shocking result also snapped Bayern's 85-game scoring streak, the longest such stretch for any team in Europe's top-five leagues during the time.