Despite dominating throughout, Bayern lacked fluency on Sunday after their top scorer's withdrawal, and two goals from Brazilian striker Raffael handed them a first loss at the Allianz Arena since April 2014.
Robben caught the eye during his time on the pitch, before a collision with Gladbach right-back Tony Jantschke resulted in him hobbling off just 24 minutes in to his comeback from a minor back problem.
And within a few minutes, Gladbach were ahead, as Manuel Neuer failed to keep out Raffael's effort.
The Dutch star was at the centre of things for the match's first real opportunity in the seventh minute, heading Juan Bernat's left-wing delivery straight at Yann Sommer.
Despite the home side's dominance, Gladbach's numbers at the back proved difficult to break down, while they also contained Robert Lewandowski to good effect.
The blow of seeing Robben forced off with an apparent hip injury was a pivotal one for Bayern, with Thomas Muller taking his place.
Within six minutes, they were trailing, as Raffael met Patrick Herrmann's precise cut-back with a low drive from 18 yards which Neuer inexplicably allowed to squirm over the goalline.
Having taken the lead, the visitors became even more compact at the back and, besides a Muller header that was played across goal and cleared by Martin Stranzl in the 42nd minute, Bayern crafted little before the break.
The second half followed a similar pattern, as Gladbach absorbed Bayern's pressure brilliantly, preventing Pep Guardiola's men from creating a single opportunity in the 15 minutes that followed the interval.
Bayern's first chance of the second period arrived in the 63rd minute, as Muller met Rafinha's cross and sent a volley at Sommer.
They were not to be denied next time, though, as Christoph Kramer raided up the right flank and crossed into the centre, where Raffael was waiting to squeeze a first-time effort under Neuer, moving Lucien Favre's men up to third.
