Champions City moved within four points of the Premier League leaders with a tense 2-1 win over the Reds in a frantic clash they could not afford to lose on Thursday.
Liverpool had arrived at the Etihad Stadium unbeaten and on a run of nine successive league wins but City are now firmly back in the picture and Silva is relishing the challenge.
"We knew that if we lost it would almost be over because 10 points is a lot," said the Portuguese.
"Now four points we believe it is possible, knowing the Premier League is a very tough competition. The pressure was a little bit more onto us. We are happy with that, it is good. We are used to that kind of pressure.
"Happily the performance from everyone was good and we got three very important points against a fantastic team."
The pace of the game was unrelenting but City prevailed thanks to superbly taken goals from Sergio Aguero and Leroy Sane either side of a Roberto Firmino equaliser.
"I think it was very good game," said Silva. "Both teams were very intense, wanted to try to play football and win the game. Happily we won it."
City now appear to be back on track after three defeats in four league games in December but the title race now takes a back seat to domestic knockout football.
City host Rotherham in the third round of the FA Cup on Sunday before Burton visit the Etihad for the first leg of their League Cup semi-final on Wednesday.
City are next in Premier League action when they host Wolves on January 14.
"Obviously when you win against the team that is top of the league it means a lot, for the confidence it is always good to beat them," Silva said.
"But it is just a game. If we lose the next one this doesn't mean anything. On to the next one and try to keep playing the same way as we have the last two games and try to not let happen what happened to us a few weeks back."