There were fears Kimmich was set for a long spell on the sidelines after he appeared to overextend his knee while fouling Erling Haaland in Sunday's (AEDT) Klassiker.
The versatile midfielder had to be helped off the pitch and, while refusing to comment on the nature or extent of the issue, coach Hansi Flick acknowledged afterwards Kimmich would be a big miss.
But Bayern revealed on Monday (AEDT) the player had suffered a right lateral meniscus injury and had already undergone successful surgery.
The ailment is still going to keep Kimmich out of action but, according to Bayern's medical staff, only until January.
Sporting director Hasan Salihamidzic said: "We are happy that Joshua will probably be available to us again in a few weeks.
"We will give him the best possible support during his rehab."
Kimmich has started all but three of Bayern's Bundesliga matches since the start of last season, although they have a trio of wins without the Germany international in the XI, scoring 12 times and conceding just once.
He has a goal and four assists in six league games this term, creating 12 chances in all.
Kimmich has completed 89.8 per cent of his Bundesliga passes this season – 87.1 per cent in the opposition half – and has won possession on 50 occasions, although he has lost the ball 85 times.