Carlos Tevez provided the gamewinner with a diving header by in the second half while Roma missed chances included a first-half penalty kick drilled over the crossbar by Daniele de Rossi.
"We had our moments in the first 25 minutes when their goalkeeper made four fantastic saves," said Manchester United coach Alex Ferguson. "But obviously the penalty would have made a big difference."
Man U faces Barcelona in the semifinals.
"I don't know if they will win [the title] but they are certainly well equipped to," said Roma coach Luciano Spalletti. "They have huge potential as a team and in terms of individual players."