The 33-year-old is in full training and could boost the club's attacking options in the upcoming fixtures, as Wolves search for some attacking inspiration.
Costa is known for his fiery approach to the game and Lage hopes his intensity will add a different dimension to Wolves' games.
"What I want is to have quality players that do everything they can to help the team win the game," Lage said.
"With his personality, he is that kind of player. That's why he won what he won at Chelsea and what he won at Atletico Madrid.
"It's so hard to be a player and a manager, sometimes I'm here and show a face that I'm not showing in training or at a game. We have time to relax.
"If you call Diego Costa here, in five minutes you'll be laughing because he has that personality.
"But for sure, if you play one-on-one against him he will break your leg if he needs to, to win the game. This is the kind of profile he has.
"He needs to be there because the Premier League is so hard to play in. Of course, breaking the leg is a joke, but what can we see in every game? Competition, duels, physical contact, intensity.
"It's emotional and when you go you need to go with that emotion and intensity to do everything to win the game. If you want to jump for the ball in the air, you can't think if something will happen, you have to jump for the duel.
"These are the kind of things he offers to the team."
Defender Nelson Semedo echoed his manager's thoughts and believes Costa will bring added 'fight' to the Wolves squad – after playing against him when both players were playing in Spain's top flight.
“It was a good signing, a very good signing because he’s a very good striker, he’s very experienced and he has already played in the Premier League," Semedo said.
"He knows the league, he knows how to fight the defenders here and we know the type of player he is. He likes to fight for every ball and he is also very good with the ball, so he can help us a lot.
“He likes to fight for every ball and he goes with everything. Sometimes you can get an elbow in the face or some stuff like that, but sometimes you have to do it to win the ball. I faced him a few times in Spain and I definitely prefer to have him in my team than playing against him.”