The 25-year-old joined the Blues for £32 million earlier in the summer following a prolific season in La Liga, in which he bagged 27 goals as Atletico Madrid won La Liga.
Now Costa, who has been joined by Atletico team-mates Thibaut Courtois and Filipe Luis in west London, is aiming to enjoy back-to-back successes by helping his new club to their first league win since the 2009-10 campaign.
“I’d say to the fans that you need to be confident that I personally will fight for this club to make sure we’re in contention to win as many trophies as possible,” he told Chelsea TV at the club’s pre-season training camp in Austria.
“I’m part of a great group of players here and we will do everything we can to give the fans what they want: and that’s titles.”
Chelsea are touring Europe before their opening game of the season against Burnley on August 18, and face Austrian side Wolfsberger on Wednesday evening.