The Brazil-born striker has long been linked with a switch to Stamford Bridge and the clubs agreed a fee early this month, with Chelsea meeting his buy-out clause reported to be worth £32million.
A statement on chelseafc.com read: "Chelsea Football Club is delighted to announce Diego Costa has today completed his transfer from Atletico Madrid by signing a five-year contract.
"He will join up with the first team for our pre-season European training camp next week."
Costa joins former Arsenal and Barcelona midfielder Cesc Fabregas in moving from the Primera Division to Stamford Bridge this summer.
His goals helped Atletico claim a first Spanish league title in 18 years last season and he helped the Madrid side to the Champions League final, beating Chelsea in the semi-final en route to Lisbon, where Real Madrid won the European Cup.
Costa said on chelseafc.com: "I am very happy to sign for Chelsea.