City snatched a stoppage-time victory over the managerless Swans at the Etihad Stadium as Kelechi Iheanacho deflected in a Yaya Toure strike.
It had look like Swansea would claim a point after Bafetimbi Gomis struck in the 90th minute to cancel out a first-half header from his former team-mate Wilfried Bony.
City had been sluggish throughout, with Pellegrini believing his side were still feeling the effects of Tuesday's draining Champions League win over Borussia Monchengladbach.
Pellegrini said: "We really didn't play well during the whole game
I think Swansea played a very good game.
"I knew three of four players were not 100 per cent recovered and when you have an important match as we did in the week, to restart the motivation is not easy.
"But it is three important points
Maybe we didn't deserve to have the points today but I think these points maybe replace the points we lost here against West Ham, when we had 18 or 20 chances to score."
Pellegrini felt Toure was City's outstanding player
The Ivorian provided energy and drive throughout and it was his determination that brought about the winner.
Pellegrini said: "I think Yaya was one of the best players in this game
In the first half and the second he was the player who tried to create different things.
"He has just come back in from an injury to his calf and he played the whole game in the Champions League and the whole game today
The goal was a reward.
"But the most important thing is we won
It doesn't matter if it is Yaya's goal or Kelechi's goal."
Swansea had troubled City in the second half with Gylfi Sigurdsson particularly impressive, forcing saves from Joe Hart and having an effort disallowed after a foul by Gomis.
It was a battling display by the Swans just three days after Garry Monk was sacked as manager following a dismal run of form
The delight on their bench was obvious as Gomis appeared to have earned them a creditable draw but that joy soon turned to agony.
Caretaker boss Alan Curtis said: "I felt we deserved something out of the game and thought we had got something when Bafi Gomis scored that equaliser, but that probably shows the difference between teams at the top of the table and teams where we are at the moment.
"When you're lacking a bit of confidence the luck tends to go against you but we can look at each other at the end of the game knowing everyone had given everything they had.
"We showed today the quality is there
We have got good players and I thought we were excellent today
So many people had written us off.
"We are bitterly disappointed to come away empty-handed but the performance, that is the standard
We have to maintain that and if we do, I am 100 percent confident we will get out of trouble."
Source : PA
Source: PA