The former Stoke City boss took over the Eagles towards the end of November when the club were at the bottom of the table having taken seven points from 12 games.
Now, the club are seven points adrift of the relegation zone in the Premier League table, with six games left to play.
"It wouldn't surprise me if Tony picked up something," Lambert told reporters. "He did brilliantly when he was at Stoke, and he's doing really well now at Palace.
"I get on pretty well with him, and you know what you get with Tony's sides, that it's going to be a hard game.
"He's a squad of good, honest pros who have picked up a couple of good results lately that has given them a fighting chance."
Palace will host Villa at Selhurst Park on Saturday.