The 25-year-old, who cost £32million this summer, is currently part of the Blues pre-season tour of Europe, where they play Austrian side RZ Pellets on Wednesday night.
Costa, who last season helped Atletico to their first La Liga title in 18 years and to the final of the Champions League, is not part of that Chelsea side but he told his club's media channel he will give his all for his new club this season.
"I'd say to the fans that you need to be confident that I personally will fight for this club to make sure we are in contention to win as many trophies as possible," Costa said.
“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."
The Brazil-born Spanish international had an unfortunate World Cup, knocked out with his side in the group stage after a poor showing not only from him but from the rest of what were the reigning world champions from 2010.
Costa is expected to be part of Jose Mourinho's pre-season fixtures when Chelsea ramp up their preparations against Vitesse Arnhem, Werder Bremen and then Ferencvaros.
Chelsea's opening fixture of the Premier League season sees them travel away to Burnley on Monday Night Football.
You can watch Chelsea v Burnley live on Sky Sports 1HD from 7pm on Monday August 18.