Robben, who opened the scoring in the third minute, was carried from the field 10 minutes later after being caught on the knee by the studs of Augsburg goalkeeper Marwin Hitz.
"Arjen has a deep cut, right into his knee joint," said club doctor Hans-Wilhelm Muller-Wohlfahrt after tests were carried out on the 29-year-old on Thursday morning.
"A stud put a lot of pressure on the femur in his right leg and there is a big build-up of blood."
As a result, Robben will miss the rest of 2013, including the Club World Cup in Morocco.
"Of course I'm bitterly disappointed," Robben said on Bayern's website. "I was in such good form and was looking forward to every game I was playing, and now this happens.
"But I'll come back fit and will then help the team reach the goals that we set ourselves before the start of the season."
Robben will hope to be fit again for the start of the second half of the season in late January, but until then, Bayern will have to play their final two Bundesliga fixtures of 2013 and their last Champions League game against Manchester City without him.
Franck Ribery also remains a doubt for those games due to injury.
"We're incredibly sorry that Arjen got injured so badly in Augsburg," said Bayern's director of sport Matthias Sammer.
"Arjen was probably in the form of his life right now and been decisive in so many matches so to lose him now is bitter, but he will get all the help he needs from us and will be back for the second half of the season."