Costa, who represented Spain in their terrible World Cup campaign, had a buy-out clause of £32 million and will become Chelsea's second big signing of the summer after the capture of Cesc Fabregas from Barcelona.
A club statement read: "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."
Costa was the inspiration behind Atleti's shock Liga triumph last season, the Brazilian-born forward scoring 27 times in 35 appearances.
The 25-year-old rejoined Atletico from Valladolid in 2010 having previously spent three loan spells away from the club before departing on a permanent basis in 2009.
In June, Chelsea boss Jose Mourinho said: "Costa is a player who I want to have in my team. I've asked my club for him and I have total confidence that we're going to achieve it and he will end up signing for Chelsea."
- Sports & Recreation
- Chelsea Football Club
- Diego Costa
- Atletico Madrid
- Cesc Fabregas