The two clubs announced on Thursday they had reached an agreement for Costa's transfer, with the terms of the deal understood to be worth an initial £48.4million with a further £8.8m in possible add-ons.
Costa has long tried to engineer a move back to Atletico, where he spent two separate spells earlier in his career, winning the Spanish league in 2014.
Though he will have to wait until January to return to competitive action Costa, whose £150,000-a-week wages are being covered by Atletico before they register him in the new year, is thrilled to be reunited with his former club.
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."
Costa may have two Premier League titles and 59 goals to his name as a Chelsea player but his final months at Stamford Bridge were defined by an increasingly acrimonious relationship with Antonio Conte.
The Chelsea boss said earlier this season that he decided to move on from Costa in January after a bust-up between player and manager over an £80m offer from Chinese Super League side Tianjin Quanjian, which the Blues refused to accept.
Costa claimed over the summer that he had been sent a text from Conte telling him he was no longer needed at the club and the Spanish international spent the final weeks of his Chelsea career in exile in Brazil.
He said of his acrimonious exit: "It's not the way I wanted it leave, ever. I have special affection for the people at Chelsea. I'm not a person who is going to smear the image. They know the person I am.
"I wanted to return (to Atletico). I'm not going to be ungrateful to Chelsea, where I was also very happy, playing in a team that was great. Atletico is my home and I'm here."
Costa will undergo double and perhaps triple sessions in a bid to recover his match sharpness and admitted it was not the most inticing aspect of his return to Spain.
"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!"