Chelsea agree to sign Costa from Atletico
The Spain international had been frequently linked with a move to Stamford Bridge following an impressive campaign with the Liga champions.
And Chelsea confirmed on Tuesday that they had met the buy-out clause for the 25-year-old.
"Chelsea Football Club can confirm an agreement has been reached with Atletico Madrid for the transfer of Diego Costa, with Chelsea meeting the buy-out clause for the Spanish international," read a statement on the Premier League club's website.
Costa, a member of the Spain squad who crashed out of the FIFA World Cup in the group stages, scored 36 goals for Atletico last season, helping the club to domestic glory and the final of the UEFA Champions League.
He becomes the second high-profile signing of the close-season from La Liga for Jose Mourinho, who also acted quickly to bring in Cesc Fabregas from Barcelona.
Costa will be tasked with helping to improve a Chelsea frontline that was often bemoaned by Mourinho during his first season back at the club.
Samuel Eto'o, Fernando Torres and Demba Ba all faced criticism from Mourinho last term - with the Portuguese prefering to name Andre Schurrle as a central striker at times.
Mourinho told Marca last month: "Costa is a player I want in my team.
"I've asked the club to bring him in and I am extremely confident that we are going to get him and he'll end up signing for Chelsea."
The length of Costa's contract at his new club has yet to be revealed, with Chelsea merely confirming that a deal to sign the player has been finalised.