The deal had been in the pipeline for a number of months, and the clubs agreed a fee earlier in July, with the move now confirmed by the Premier League club.
The fee for the Brazil-born player, who represented Spain in the World Cup, has not been revealed but it is believed to be in the region of £32million.
"Chelsea Football Club is delighted to announce Diego Costa has today completed his transfer from Atletico Madrid by signing a five-year contract," read a statement on the club’s official website.
"He will join up with the first team for our pre-season European training camp next week."
Costa was delighted to see the move finally completed and is keen to test himself with a club he faced as an Atletico player last season.
Costa scored the second goal in the Spanish side’s 3-1 win at Stamford Bridge in their UEFA Champions League semi-final second leg match last season to help secure their final place in the final against rivals Real Madrid.
The 25-year-old also scored 35 goals last season to help Atletico win their first Spanish league title in 18 years and he is now ready to begin his fresh challenge in the Premier League.
"I am very happy to sign for Chelsea,” said Costa.
"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.
"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."
Watch Chelsea's opening game of the new season away to Burnley on Monday August 18 on Sky Sports 1.