The €43m signing dribbled past two players before scoring, while Andre Schurrle, Oscar and Ramires featuring for Jose Mourinho's side after returning to training earlier in August
Chelsea faced both clubs in two 45-minute matches on Friday evening at the Turkish charity tournament in aid of the Soma mining disaster.
Eden Hazard was given his first start in pre-season against Fenerbahce, while Andre Schurrle and Oscar also started following their return to training earlier this week. Ramires began the game as a substitute but replaced Schurrle after 33 minutes.
Petr Cech was preferred to Thibaut Courtois in both games and the 32-year-old pulled off a fine save from Alper Potuk's close-range header.
Chelsea opened the scoring in the 24th minute through Costa's impressive individual goal. The €43 million arrival from Atletico Madrid picked up the ball midway in Fenerbahce's half and slalomed between two defenders before he dispatched a cool, side-footed finish past Volkan Demirel.
Branislav Ivanovic doubled Chelsea's lead six minutes later as he pounced on Demirel’s save from Schurrle’s curled free-kick to steer the ball home.
Chelsea's second game saw a defeat as Ersan Gulum scored with two minutes to go to crown Besiktas the tournament's overall winners.
Recently departed Blues striker Demba Ba came on after 15 minutes to face his former side but, in a game slow to ignite, returning legend Didier Drogba came close soon afterwards with a free kick that drew a good save from Cenk Gonen.
Ba later came close to netting a parting shot, with Petr Cech deflecting his late effort out for a corner.
It was from that set-piece, however, that Besiktas got their goal, with Ersan finding the delivery right at his feet to fire past Cech.
Chelsea got World Cup contingent Andre Schurrle, Oscar and Ramires all onto the pitch for their first taste of pre-season action but otherwise had little else about which to be excited after a dour 45 minutes.