After a bright first half from the visitors at Etihad Stadium, Stevan Jovetic stole in to smash home from close range four minutes before half-time.
The Montenegrin then made it 2-0 shortly after the break when he fired in Samir Nasri’s cutback.
I felt that we needed to be a bit quicker with our actions
Substitute Sergio Agüero made it three when he scored with his first touch in the 69th minute.
Liverpool then netted a late consolation when Pablo Zabaleta inadvertently bundled home after Rickie Lambert’s effort.
“Up until 40 minutes we were the best team and then we’ve just switched off in the final period of that half and then all of a sudden we’re 1-0 down,” Rodgers told Sky Sports.
“The second goal killed us a bit and 2-0 was difficult for us. Credit to the players, they kept going. We had a good reaction, we kept going and went to 3-1 and we still created chances.
“I felt that we needed to be a bit quicker with our actions in the final third and be a bit more dynamic.
“It’s a difficult place to come. We’ll continue working as it goes on. It just wasn’t our night.
“We’re three points better off than we were last season in the corresponding fixtures [City and Southampton]. There’s a hell of a long way to go.”