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. City and Mancini received considerable criticism after suffering defeat to Ajax in the Champions League in midweek and the manager felt that game had taken much out of his players.
The Italian 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."
He added: "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."
City's injuries are now mounting. Already without Maicon, Pablo Zabaleta, Javi Garcia and Jack Rodwell, City lost Aleksandar Kolarov and Micah Richards during the game.
The Swans frustrated the champions throughout a game that had 12 minutes of injury time in the second half after stoppages for treatment to Richards and Michel Vorm. The visitors also had two good chances to score while the game was still locked at 0-0 but Joe Hart saved from Michu on both occasions.
Manager Michael Laudrup said: "We have to say, both times, Joe Hart made some great saves.
"I really think, coming here, playing against the champions away, when they only create one clear chance in 90 minutes and you create two or three, you can think you deserved a little more.
"The great teams just take those few chances and convert them into goals. I want to only see the positive things today because there are so many."
Source: PA
Source: PA