Former England midfielder Cole, 32, has joined Villa on a two-year contract after being released by West Ham at the end of last season.
Lambert told his club’s official website: "Joe's not in it for the money, he simply wants to play and prove that he's still a top professional in the Barclays Premier League, which he's always been.
"He'll thrive on the responsibility here. He's someone who has always stepped up to the plate. That's inherent in his character and he's the type of character we need.
"I'm delighted we've been able to bring him in to join the group. He'll enhance the group and he'll bring out a lot from players around him because he'll demand that.
"He's a humble lad in spite of everything he's achieved in the game and he's hungry as well. Money was never a factor in him coming here. He simply wants to achieve and try to win things, that's what drives him.
"For us, he's got a lot to offer - he's an intelligent footballer and his attitude is first-class.
"When we sat down and talked I was very impressed by him. He's a guy who has played at the top for many years, been capped 56 times for England which is incredible but he still has an awful lot to offer and he wants to prove himself still.
"This is a really good signing for us. If the fire wasn't still burning in him, which is the phrase he's used himself, he would have gone overseas and been content with what he's achieved.
"But that's not him. Players will learn from Joe Cole, as they will from Philippe Senderos who joined us last week. I'm looking forward to both of them being part of the group and the other players will be excited by them joining, too."