Brazil international Costa is already ruled out of Juve's next four Serie A matches after he was sent off for spitting at Sassuolo forward Federico Di Francesco in Sunday's 2-1 win.
He featured from the bench in the second half against Valencia, but towards the end he was forced off, needing support as he hobbled off the pitch.
Following examinations, Juve confirmed on Friday that Costa's injury amounts to a sprained ankle and "a distortion to the adductors of the right thigh".
Although Juve have not given a timescale for his return, reports in Italy claim the 28-year-old could be out for as long as a month.
If that proves to be the case, Costa will be unavailable for the Champions League game against Young Boys on October 2, while he could also be a doubt to face Genoa in Serie A 18 days later following the conclusion of his domestic suspension.