The 24-year-old will miss next weekend's Premier League game against Crystal Palace, with Chris Smalling already out of that fixture after he was sent off against City.
Asked how long the Argentine will be out for, United boss Louis van Gaal said: "We will have to wait and see."
Rojo joins fellow defender Phil Jones (shin splints) on the sidelines.
Centre-back Jonny Evans has also not played since injuring his ankle in the 5-3 defeat by Leicester on 21 September.
United finished the game against City with midfielder Michael Carrick and 19-year-old defender Paddy McNair in defence.
"I have seen good performances from Carrick and McNair, but it's a disaster for us," Van Gaal added.
"So many defenders are injured and you have to play each week with another defence.
"Despite the result against Manchester City, I think we are very close. When you see what we have done with 10 players against 11 you can be proud as a coach."
Manchester United defender Marcos Rojo suffered a dislocated shoulder in Sunday's 1-0 defeat by Manchester City.