The Dutch winger suffered a severe cut to the knee following a collision with Augsburg keeper Marvin Hitz. Robben received immediate treatment but was stretchered off the pitch after just 13 minutes of the German Cup clash – his knee gushing with blood.
"Robben has sustained a deep laceration extending into the knee joint," Bayern doctor Hans-Wilhelm Muller-Wohlfahrt told the club's website.
"The impact of one of the studs on the thigh bone in the right leg has caused extensive haemorrhaging."
The 29-year-old had earlier opened the scoring, notching his fifth goal in four games before suffering the nasty injury.
Following the news, Robben said: "Of course I'm really disappointed. I've been feeling great and have been enjoying playing every game with the team.
"But I will return to fitness and will join the team in tackling the goals we set ourselves for the season."
Bundesliga leaders Bayern confirmed the news on their Twitter account.