Bayer Leverkusen, runners-up last year, also paid for lapses as the back as they lost 2-0 at Mainz.
Igor de Camargo scored the only goal in the 62nd minute for Monchengladbach after Bayern goalkeeper Manuel Neuer and defender Jerome Boateng got into a terrible mix-up over a long ball forward by Roel Brouwers.
Neuer, signed from Schalke 04 amid protests from a group of Bayern fans angry at the purchase of a former rival, rushed out to try and clear the ball and former Manchester City defender Boateng left it for him.
But Brazilian-born Belgian international De Camargo nipped in to head the ball over the goalkeeper from the edge of the area and give Monchengladbach, who escaped relegation by the skin of their teeth last season, a shock win.
"Gladbach were very well-organised. They built a spider's web in which we always got caught," said Bayern coach Jupp Heynckes, starting his third stint in charge of the Bavarians.
"The goal came from nowhere, it was a mis-understanding between Boateng and Neuer."
Bayern, who fired Louis van Gaal and replaced him with Heynckes in their bid to improve on last season's disappointing third place, dominated the first half without creating any real chances.
The best opening fell to Monchengladbach when Neuer did well to tip over a cheeky chip from Venezuela midfielder Juan Arango.
Mario Gomez headed against the post after the interval and the Bavarians, who started with Arjen Robben and brought on Franck Ribery after an hour with their two top players recovered from pre-season injuries.
After falling behind, Bayern threw everyone forward and Thomas Muller had a goal disallowed for offside but they looked toothless in attack.
Leverkusen fell behind after 32 minutes following a dreadful mistake by Fabian Giefer, whose miscued clearance went straight to Sami Allagui who rifled home.
Four minutes from time, Omer Toprak turned the ball into his own net as he lunged to intercept Marco Caligiuri's centre.
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