Alaba was forced off during Bayern's 2-0 win over Roma in the UEFA Champions League on Wednesday - a victory that secured the German club's progression to the last 16.
Their qualification was soured somewhat by the injury to the 22-year-old, with Bayern confirming the extent of his problem on Thursday.
Posting on their official Twitter account, the Bundesliga champions announced that Alaba had suffered a "partial MCL tear" as well as a "medial meniscus injury".
"He is set for several weeks on the sidelines," Bayern added.
Alaba will now undergo surgery, with the Austrian adding to coach Pep Guardiola's lengthy injury list.
Bastian Schweinsteiger, Thiago Alcantara and Javi Martinez remain sidelined, while Claudio Pizarro was ruled out for a number of weeks due to a thigh problem on Wednesday.
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