The Barclays Premier League champions were again far from their best but moved within a point of the top after Carlos Tevez settled an otherwise poor game with a fine strike on the hour.
Mancini said: "Our performance in the first half was so-so because I think we were so tired but in the second half we played better, we had a lot of chances to score.
"In this moment, when you are in a difficult moment, after a defeat in the Champions League, it is important we start to win quickly.
"We played three days ago and Swansea played very well. They were fresh, so it was important to take these three points.
"We now have one week when we can recover and I hope we can recover a couple of (injured) players for the next game and we can work."