Jose Mourinho's Blues announced on July 1 that they had met the Brazilian-born striker's release clause and that he would join them in the coming weeks.
Costa, who represented Spain in a woeful World Cup, has now completed his move to Stamford Bridge and will join his new teammates at their training camp next week.
Chelsea announced the news on their website. A statement 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."
The 25-year-old expressed his joy at joining the London-based outfit following the announcement of deal.
"I am very happy to sign for Chelsea. Everybody knows it is a big club in a very competitive league, and I am very excited to get started in England with a fantastic coach and team-mates. Having played against Chelsea last season I know the high quality of the squad I am joining," Costa told Chelsea's website.
"I would like to thank everybody at Atletico who made me into the player I am, it was an incredible time for me, but now I am starting a new adventure and I hope to win many trophies with Chelsea."
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