The champions surrendered a 2-0 lead for the second time in a week, after Sunday's draw against Burnley, but this time fought back to beat the Black Cats 3-2 in the Barclays Premier League.
Frank Lampard, whose short-term spell at the club was extended on the eve of the match, came off the bench to head the winner in the 73rd minute.
The former England midfielder might even have had more in a frantic finish as he brought two fine saves out of Costel Pantilimon, who also denied Jesus Navas and Fernandinho.
Pellegrini said: "Once again this team had the personality and the character to go for the third goal.
"It was very similar to what happened against Burnley but I think we didn't have any cost today.
"We went for the third goal and we had three or four more clear chances to score goals
I think in that sense Pantilimon was their best player.
"We are not trying to win to make pressure on Chelsea - we just try to win to continue playing in the way we are doing so far
We have won 29 points from the last 33
It's a very important average of points and we are going to try and continue in the same way we are doing so far.
"At the end of the season we will see which team which has more points."
The first half was goalless and dull, but the game burst into life when Yaya Toure crashed in a ferocious shot after 57 minutes and Stevan Jovetic added another.
Sunderland hit back with quickfire goals from former City players Jack Rodwell and Adam Johnson - the latter a penalty - but Lampard stole the show.
Sunderland boss Gus Poyet said: "The disappointing thing of the performance is after 2-2, the last 15 minutes, we gave the ball away too easily.
"It's Frank Lampard, we know what he does
He's going to arrive late in the box so we need to mark.
"We opened up the game completely and we needed our goalkeeper to be very good at the end to save us from conceding five or six.
"It's not surprising when you open up against a team of Man City's quality - we need to be realistic - but we gave the champions a decent game."
Lampard might not have been playing had City not reached agreement with their sister club New York City to retain his services on New Year's Eve.
The arrangement is an unusual one because of the close relationship between the two clubs but Poyet was not particularly concerned about the nature of the deal.
He said: "It looked to me last week that it was not possible (for Lampard to play) but apparently it was
How I don't know, I'm not an expert on the rules and regulations - but he made a difference, he scored the winning goal.
"For me it's clever, it's smart
It's legal, so I've got no problem with that."
Source : PA