The Argentina international seemed to hurt his knee in a challenge with Frank Acheampong and was replaced by Daley Blind midway through the first half at Old Trafford.
Both Chris Smalling and Phil Jones are currently on the sidelines, with Blind and Eric Bailly now the club's only fit senior centre-backs.
Rojo and Bailly were lining up for their sixth straight game together at the heart of the defence following on from injuries picked up by Smalling and Jones during England duty, but the 27-year-old was forced off inside 23 minutes having been caught on the right knee when attempting to cut off Acheampong's burst down Anderlecht's left.
Moments earlier he had been caught in another collision in midfield and twisted his left knee as he went to ground, forcing him to receive a period of treatment.
United have upto 10 games remaining in their congested season, and any further injuries could see Jose Mourinho forced to call up under-23 skipper Axel Tuanzebe, whose only previous first-team experience was a late run-out in the 4-0 FA Cup fourth-round win against Wigan Athletic in January.