The Blues announced at the start of July that they had reached a deal with the Spanish champions for the Brazil-born forward, although an agreement has been in place for months and Costa himself previously told Goal why he has chosen the Londoners.
Costa has won two Uefa Super Cups, the Copa del Rey and the Primera Division during his four years with Atletico, and hopes to keep up the winning habit with his new club.
"I am very happy to sign for Chelsea," he told the Blues' official website. "Everybody knows it is a big club in a very competitive league, and I am very excited to get started in England with a fantastic coach and team-mates.
"Having played against Chelsea last season I know the high quality of the squad I am joining.
"I would like to thank everybody at Atletico who made me into the player I am, it was an incredible time for me, but now I am starting a new adventure and I hope to win many trophies with Chelsea."
In Goal's interview with the Spain international, published during the World Cup, Costa revealed it has long been his dream to play in England.
"I've wanted to play in the Premier League for a while. Chelsea are one of the greatest clubs of the world so there are many factors that make you take a decision like this.
"I will have to work a lot to fit in with the style of the Premier League."