The London based club posted a communique on their official website confirming they have paid the 38 million euros ($52 million) buyout clause for the Brazilian born Spain international, who formed part of the Spanish squadthat was knocked out of the group stage at the FIFA World Cup in Brazil, reports Xinhua.
"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.
The 25-year-old Costa moved to Portugal in 2006, before joining Atletico Madrid in 2007.
He then had several loan deals and even signed for Valladolid in 2009, where his form led to Atletico buying him back before sending him on another loan to Rayo Vallecano, where he hit 10 goals in 16 appearances.
Last season, he led the Atletico attack following the sale of Radamel Falcao and responded with 27 league goals as the Madrid-based club won the BBVA Primera Liga and also reached the final of the Champions League.
It will be interesting to see how his reputation for theatrics on the pitch will be accepted in England, but there is no doubt he should add more bite to Chelsea's attack following the decision Monday not to extend the contract of veteran forward, Samuel Eto'o.