The win sees them go three points ahead of Manchester United in the Barclays Premier League title race, though Gareth Barry's second-half sending-off provided the one sour note.
Sergio Aguero, Yaya Toure and James Milner, from the penalty spot, were the scorers for City.
Mancini was particularly happy with the result, especially after his side had been in action just two days ago - a 1-0 defeat to Sunderland.
"This was a very important win," he told Sky Sports 1.
"It was a difficult game because Liverpool is a fantastic team and also because Liverpool played four days ago and we played two days ago.
"This was terrible for us."
The Italian also apologised for brandishing an imaginary yellow card on the touchline when Martin Skrtel gave away the penalty with a foul on Toure.
"In the moment maybe I did that because I was not happy with the referee, but I made a mistake," he added.
Man-of-the-match Vincent Kompany added: "This is massive for us. We were incredibly efficient and it's a great win.
"It is one of the top six teams out of the way and now we know United and the other teams have to play each other."