Robben, who has only recently returned from a back injury, was taken off after just 24 minutes following a challenge from Gladbach’s Tony Jantschke.
The Dutchman could now be a doubt for Bayern’s Champions League quarter-final first leg against Porto on April 15 with the club saying he would be out “for several weeks.”
“His injury is almost as bad as the defeat,” said Bayern Munich CEO Karl-Heinz Rummenigge.
The winger, who missed last week’s win over Werder Bremen and was a surprise inclusion in the starting line-up on Sunday, is the league’s second-top scorer with 17 goals this term.
Deespite suffering their first home defeat of the season, Bayern are still 10 points clear at the top of the Bundesliga.
“They barely got into our half before they scored their first goal and it’s not easy to break down a side who pack their defence,” said Bayern boss Pep Guardiola.
“We failed to break down their defence and they counter-attacked superbly, but we did all we could. We still have a 10-point lead, but we must fight until the end to defend our title.
“We still have eight games left and we have the chance now to regroup during the international break.”