Ryan Nelsen went close to another equaliser late on but Carlos Tevez gave the scoreline a more convincing look with his fourth goal of the campaign in injury time.
Mancini, for whom the match was his 100th in charge in the Premier League, told ESPN: "I think we played well in first half, but we have one problem at the moment - when we have a chance to score we don't score and the game can change.
"We had this problem in the first game and against Liverpool.
"In the second half we didn't play well. We were too slow but in the end it's important that we took three points."