A deal worth 75 million pounds, including transfer fee and salary, was finalized at a meeting in London less than 24 hours after Costa's penalty inspired Atletico Madrid's 3-1 Champions League win over the Blues on Wednesday.
Chelsea will pay around £35 million for the striker, although a significant portion of the transfer fee includes agents' fees.
The Europa League champion will also more than triple the wages of Costa, who is believed to earn around £50,000 per week at the Estadio Vicente Calderon. Chelsea is set to hand the 25-year-old a five-year contract worth £150,000 a week.
Representatives of the London club have held a series of meetings with Costa's camp in recent months and the latest set of talks around the two clubs' Champions League showdown has closed the deal.
Speaking in the wake of Wednesday night's win, Costa admitted that he had a few offers on the table from rival clubs but said that he would focus on Atletico's pursuit of the double before making a decision on his future.
Atletico coach Diego Simeone had paved the way in April for Costa to leave by admitting that the Spanish club was powerless to stop him being sold to a club of Chelsea's financial might.
"It will depend on him [Costa] and what he thinks is the best for his future," Simeone said. "I will support any decision he makes. It's natural that Chelsea are interested in him.
"We understand that Chelsea have huge financial power and if he wants to live off football for the rest of his life, I will have no problem in him going. Samuel Eto'o is getting older and Diego Costa could do his thing there but, for now, he is ours."
Costa emerged as Chelsea's top center forward target following an outstanding campaign for Atletico, which heads the Spanish table as well as being in the final of the Champions League. The 25-year-old has scored 36 times in 49 matches this season and his 27-goal league haul is second only to Cristiano Ronaldo (30).
The Blues were keen to trigger the Brazil-born forward's release clause in January, when he was also being closely monitored by Arsenal, but the player did not want to leave in mid-season.
The buyout figure was understood to be £30 million in January, although a slightly higher sum of £35 million will be paid to capture him in the summer window.
Arsenal dropped its interest in Costa in the winter following lukewarm scouting reports, leaving Chelsea in pole position to capture the prolific striker.
Jose Mourinho has made no attempt to hide his desire to recruit a blue chip center forward amid doubts over the futures of all three of the Blues' senior front men - Eto'o, Fernando Torres and Demba Ba. At least two of the trio are expected to leave in the summer, while Romelu Lukaku will hold talks with Chelsea chiefs when he returns from his season-long loan at Everton.
As reported by Goal, Atletico leads the chase for Torres and there is also interest in the Spaniard from Inter, Tottenham and Galatasaray. A growing possibility is that Torres could leave on a season-long loan to free up space in Chelsea's attack and on the wage bill.
Ba's representatives have already begun speaking to interested parties, while current first choice Eto'o, 33, is out of contract in the summer.