The German international, 25, suffered the injury in the first half of Saturday's 3-2 win at Borussia Dortmund which sent them top of the Bundesliga.
"Joshua Kimmich underwent successful surgery on his right lateral meniscus on Sunday evening," said Bayern in a statement.
"The FC Bayern medical department expect the midfielder to be available again in January."
Bayern board member Hasan Salihamidzic added: "We're glad that Joshua will probably be available again in a matter of weeks. We'll support him in his rehabilitation as best we can."
The lay-off will sideline him for the final three games of the Champions League group stage as well as six Bundesliga fixtures.