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.
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.
"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.
Kolarov was removed at half-time with a foot problem but of more concern was the injury to Richards, who left the field in agony wearing an oxygen mask after appearing to twist his knee.
The England defender missed the first two months of the season with an ankle injury and was only in his fourth game back. He will now be assessed in the coming days.
When asked if it was a bad injury, Mancini said: "Probably, but I hope not. I don't know. He was tired, it was the third game in 10 days for him."
City struggled in the first half but upped the tempo after the introduction of Mario Balotelli at half-time.
When asked what he had said to his players at the break, Mancini mumbled a possible joke about shooting "against three or four players".
Mancini was however clear when asked about the performance of Tevez, who scored his first goal since September 1.
Mancini said: "He played well but Carlos also played well against West Bromwich.
"He worked hard for the team. It is an important goal and I am happy for him."
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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 [Saturday] because there are so many."
Aside from the result, the only other negative was a booking for Angel Rangel for diving in the first half.
Laudrup said: "Angel got a fifth yellow card - that was really something, that was something special. I have to admit that.
"A free-kick against and it was strange. If that is diving, then we have invented a new form of diving.
"I have to ask Angel how he learned that."