Robben left on a stretcher 14 minutes into the game after a challenge with Augsburg keeper Marwin Hitz left him with a deep gash in his right knee.
The Dutch attacker scored four minutes into the 2-0 victory as Bayern progressed to the final eight of the DFB-Pokal and has been in stunning form of late, scoring in his last four Bundesliga matches.
Bayern's head coach Pep Guardiola was not sure how serious the injury was but hoped Robben would be ready for the Club World Cup in Morocco on December 17.
"I hope Arjen can help us for the Club World Cup at the latest," Guardiola said on Wednesday.
"I am really sorry for him, we didn't want that."
Bayern sporting director Matthias Sammer said that Robben would undergo scans to ascertain the severity of the injury.
"We'll only know for sure once he has been examined at the hospital," Sammer said.
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