The Spanish striker, signed from Atletico Madrid, has promised total commitment to the cause as the Blues look to win a first championship since 2009-10.
The 25-year-old joined the Blues for 40 million euros 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 teammates Thibaut Courtois and Filipe Luis in west London, is aiming to enjoy back-to-back successes by helping his new club to its first league win since the 2009-10 campaign.
MORE: Beautiful fans in Brazil | Barcelona's new bright kits | Juve's jerseys
“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 preseason 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 is touring Europe before its opening game of the season against Burnley on August 18, and faces Austrian side Wolfsberger on Wednesday.