The 25-year-old Spain international is heading to Stamford Bridge after the Blues met the 40 million euro (£32m) buy-out clause in his contract with Atletico.
The Premier League club announced the news on their official website in a short statement.
The 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 has long been linked with a move to Chelsea as manager Jose Mourinho looks to strengthen his attacking options for next season.
In a statement on Tuesday night, Atletico said Costa had sought a move to Chelsea but the club had insisted on receiving his full buy-out clause.
"Diego Costa communicated at the end of last season his will to be transferred to Chelsea and today the two clubs have reached an agreement for which the London club must pay the full amount set as the buy-out clause in the contract between the player and the club," the statement said.
He spearheaded Atletico’s attack in a memorable 2013/14 campaign in which they won La Liga and reached the Champions League final under Argentine manager Diego Simeone.
Costa scored 36 goals in all competitions before heading to his native Brazil for the World Cup with Spain, although he failed to make an impact as the defending champions made an early exit from the tournament.
You'll be able to watch Diego Costa in Champions League action next season on Sky Sports 5 - the new home of European football.