Paolo Di Canio has helped the Black Cats move six points clear of the bottom three but Lambert reckons they cannot afford to rest on their laurels.
Lambert said: "A lot can be made of a new manager coming in and it's up to us to combat that and go and try to win.
"You don't know if it's worked, you have to wait and see, they're still bang in it. I think Paolo would say that himself.
"You can talk about new manager syndrome or whatever but it's the same group of players that were there before.
"They got a result against Everton that was really good the way Everton have been playing but we're at home and the onus is on us to make the running."
Lambert also played down suggestions that there would be any ill-will between the two dug-outs after Di Canio - then managing Swindon Town - appeared to taunt Villa fans about relegation after a League Cup game earlier this season.
"I didn't know anything about it," he added. "I think I've got more things to worry about. I don't really get wound up by that sort of thing.
"The lads that go out and play, they're the most important people along with the people that come and watch it."