Alaba was on the end of a wild early challenge from Serge Aurier and had to be substituted at half-time of the UEFA Champions League clash, complaining of severe pain.
Bayern initially announced the defender had sustained only "a heavy bruise", reporting his ribs had not been broken.
But Niko Kovac confirmed on Saturday (AEST) that Alaba, along with fellow defender Lucas Hernandez, would miss Saturday's league clash with Hoffenheim.
An additional update followed as Bayern revealed a further assessment from club doctor Hans-Wilhelm Muller-Wohlfahrt had identified the fracture.
A short statement gave no indication of a return date for Alaba, stating only that the left-back would be out "for the time being".
Alaba had been out with a thigh strain prior to the big win at Spurs, sitting on the bench for Bayern's victory over Paderborn in the Bundesliga last weekend.