City came from behind to knock the Reds off top spot and climb to second with a 2-1 success at the Etihad Stadium on Boxing Day.
Captain Vincent Kompany and Alvaro Negredo both netted before the interval after Philippe Coutinho had given the visitors a 24th-minute lead.
City now trail new leaders Arsenal by a point with Liverpool down to fourth.
Pellegrini said: "It was very important because Liverpool were the best team of the Premier League.
"They were top of the table before the match so if they won, they would be four points ahead of our team.
"I am very happy with the response because it is not easy when you face such a difficult team, especially on the counter-attack when you are losing 1-0.
"It is very easy to leave space for them. I think we defended very well."
In-form Liverpool talisman Luis Suarez was a constant thorn in the City side but Pellegrini felt his men did well to frustrate the 19-goal striker.
The Chilean said: "They did very well, especially against Suarez.
"He is a great striker. He is in his best moment so it was very difficult for our team not to lose one second in the whole match."
City had some moments of alarm, not least when Raheem Sterling put the ball in the net only to be incorrectly and controversially ruled offside.
Sterling later put a good chance over, Suarez was denied a late penalty and Joe Hart saved well from Coutinho and Jordan Henderson.
Pellegrini was particularly pleased with Hart's contribution having recently restored the England international to the side.
"I am very pleased with Joe - not only now but I think in all the games he played in the last month he did very well," the City boss said.
"I thought it was good for Joe to have a rest. He was not in his best moment.
"He has worked very hard in the last month and a half. Now it was his normal performance. He is the best goalkeeper in England."
City, who are back in action against Crystal Palace on Saturday, are now closer to fulfilling Pellegrini's aim of reaching top spot by new year.
He said: "At least we are near. We will see what happens but this is just half of the Premier League (gone).
"We have all the second half and that will be very close - five or six teams (competing) for the title."
Pellegrini reiterated his irritation with the fixture scheduling that sees City facing a quicker turnaround than a number of their title rivals this week.
He said: "All the teams play exactly the same (number of) games, but I think we must have at least 72 hours from this game to another."
"All the other teams at the top of the table, apart from Manchester United, play 72 hours after."