Chelsea met the buyout clause in the 25-year-old’s contract with Atletico and he will now negotiate personal terms.
The Brazil-born forward is in Spain’s World Cup squad.
Chelsea manager Jose Mourinho said in May that the addition of a “killer” strikerwould be a key priority during the summer.
The London club scored 71 goals in the Premier League this season, 31 fewer than champions Manchester City, who finished four points ahead of them.
Chelsea strikers Samuel Eto’o, Fernando Torres and Demba Ba contributed a total of 19 of those in the league, with 33-year-old Eto’o, now a free agent, scoring nine of them.
“We have to try to win as a team, to improve as a team, but also add the attacking player with that killer instinct,” Mourinho said last month.
Mourinho will hope that Costa will go some way towards addressing that problem having enjoyed a prolific campaign with Atletico.
Costa scored 27 goals in La Liga and only finished third behind Real Madrid’s Cristiano Ronaldo (31) and Barcelona’s Lionel Messi (28) on the list of top goalscorers.
The striker also scored against Chelsea from the penalty spot in the semi-finals of the Champions League, one of his eight goals in the tournament, as Atletico came from behind to win 3-1 at Stamford Bridge.
Costa was born and raised in Brazil and made two friendly appearances for them last year before he became a naturalised Spaniard.