The 25-year-old has long been linked with a move to Stamford Bridge, and the switch has now been confirmed by the Blues
As reported by Goal in early May, the striker's move to Stamford Bridge was tied up following the Blues' Champions League semi-final exit to Atletico Madrid.
Chelsea will pay around €44m for the striker, although a significant portion of the transfer fee includes fees paid to agents.
The Brazil-born Spain international will also triple his wages at Stamford Bridge, with Chelsea are set to hand the 25-year-old a five-year contract worth €194,000 a week.
The move has long been mooted, with Costa himself confirming the agreement exclusively to Goal in June: "Why would I have chosen to play for Chelsea? It is a thing that is about to happen
"Things are going in a direction and, if they keep going as they are going, it is the team I will play for in the next season.
"I've wanted to play in the Premier League for a while. Chelsea are one of the greatest clubs of the world so there are many factors that make you take a decision like this."
Costa becomes Chelsea's second signing of the summer following the capture of Cesc Fabregas from Arsenal.