Suarez replaced Costa in Atletico's 6-1 win over Granada, and him admitted the pair of them can fight and bite in reference to the biting incidents that the Uruguayan has been involved in in the past.
Costa said: "He is very good, one of us can do the fighting and the other can do the biting."
The Spanish striker admits that he couldn't believe his club's La Liga rivals Barcelona decided his new teammate was surplus to requirements.
He added: "We are very lucky to have Luis, I don't understand how Barca could have let him leave.
"He's going to help us with lots of goals as well as his warrior spirit and his hunger to win trophies. I hope he can help us win another trophy for our fans, but we need to take each game as it comes."
Costa has struggled since rejoining Atletico from Chelsea in 2018 and revealed that he has told the club that he is prepared to leave if he is no longer wanted.
He explained: "I don't want to fight with them and become a drain on the club, I've spoken to the club and the coach and told them I'm ready to leave if they think I should go.
"But if I stay I'm going to fight to play and I hope I don't get injured so I can get a good run of games. I have to make the most of my opportunities. Anything can happen this season, the important thing is to be ready and score goals."