The visitors had to overcome being denied a clear penalty when Sergio Aguero was fouled by Jose Fonte early on, with the striker booked for simulation instead, but took the lead through Yaya Toure's 51st-minute strike.
Eliaquim Mangala was then sent off for a second bookable offence on 74 minutes but City still managed to add further goals from Lampard and Gael Clichy on the break late on.
"We knew we were coming up against a good team in form, but you could see from the outset our attitude was bang on," Lampard told Sky Sports.
"The football was fabulous for the first 20 minutes and we should have had a penalty, but the spirit was there and that was what won us the game.
"We're disappointed with where we are [six points behind Premier League leaders Chelsea], because we've dropped some points that a squad as good as this shouldn't, but we have to stay in the hunt and keep winning."
Clichy produced a fine finish after being set up by Aguero to score his first goal for City.
He added: "It was a nice feeling, but I'm not after the goals. The performance from the team was unbelievable. We controlled the game.
"This is a tough place to come, so to keep a clean sheet and get three points is a perfect day for us."
Manuel Pellegrini has suggested he would like Lampard to stay on past his proposed period with City, rather than leave to join New York City FC in January.
The midfielder reacted to questions over his future by stating: "I haven't spoken to them on that front at all, so I don't know and I'm not looking that far ahead.
"It's nice that they are saying that, it's much better than them saying they can't wait for me to go, but we'll see what happens."
