Speaking after a 6-1 win against Granada in La Liga, a game in which Costa scored the opener before being replaced by Suarez on the 71st minute with the score 3-0, he said: "It's great. One bites and the other kicks."
Suarez came on for his Atletico debut and scored two and set up another.
The Uruguay international moved to Atletico from Barcelona last week on a free transfer, although Barca could make up to €6 million on the move based on a number of performance-related variables.
Costa was referencing his reputation as a feisty player who would always get in tangles with opposition defences.
Meanwhile, Suarez is notorious for biting PSV Eindhoven midfielder Otman Bakkal while with Ajax in 2010. He committed similar offences when he bit Chelsea defender Branislav Ivanovic during his time as a Liverpool player in 2013 and Italy defender Giorgio Chiellini while playing for Uruguay at the 2014 World Cup.
Costa added that he could barely believe his side's luck in landing Suarez, who scored 198 goals in six seasons with Barca but was deemed surplus to requirements by new coach Ronald Koeman.
"We are very lucky to have Luis, I don't understand how Barca could have let him leave," Costa said.
"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."
Suarez added of his debut: "I'm very happy, first with my official debut and, second with the victory which is what we wanted.
"I'm very happy. I am happy because when I came on were already in control and it was great to enjoy playing in such a great stadium and with great players.
"I believe that what is important and gives you confidence is how the team welcomes you. This is fundamental and very important. This is what I felt here in the game. I want to thank everyone for the support -- this will be very important in helping us achieve our goals this season."
Information from Reuters was used in this report.