On Friday afternoon landed in Madrid from Sao Paulo to undergo a medical with Diego Simeone's side and finalise his contract.
In quotes reported by Marca, Costa told reporters at Barajas airport: "I'm very happy, It's taken a lot of time but it has a happy ending. Now I'm waiting to see my team-mates and train. I'm looking forward to giving my best as always, and hopefully things go well.
"It always makes you a little nervous when time is passing by, but (Atletico chief executive) Miguel Angel Gil has made a great effort and I'm grateful to him, like everybody at the club. I'm very happy and will try to show my thanks in some way."
On the state of his fitness, Costa insists he's not in such bad shape.
"I'm not as bad as some say and here we have El Profe Ortega (Atletico's fitness coach Oscar Ortega) who is going to have me in shape. The scales don't scare me, his training sessions scare me!"