The 25-year-old Spanish international has penned a five-year deal at Stamford Bridge for an undisclosed fee.
However, it is understood that Costa has joined the Blues for around £32m.
Chelsea and Atletico agreed the move at the start of the month before finalising it on Tuesday afternoon.
"I am very happy to sign for Chelsea. 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 teammates," Costa told Chelsea's official website.
"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."
Costa, who joined Atletico Madrid from Real Valladolid in 2010, scored 35 goals in 44 games last season.