The friendly tournament, being held at Fenerbahce's Sukru Saracoglu Stadium in aid of the Soma mining disaster, followed a league format with the three sides squaring off in 45-minute matches.
Chelsea manager Jose Mourinho named a strong starting XI for the 2-0 win over the hosts, with the likes of Andre Schurrle and Oscar starting after rejoining the squad following their World Cup exploits.
Costa - playing as the lone striker - opened the scoring with a fine solo goal, jinking his way past two defenders before rifling an effort in the bottom corner.
It was the Spain international's second goal of pre-season, after the close-season signing from Atletico Madrid also netted in the 2-1 win over Olimpija on his debut last month.
Branislav Ivanovic extended the lead on the half hour when he reacted quickest to Schurrle's parried free-kick.
However, Chelsea were beaten 1-0 in the following match with Besiktas - the Turkish side scoring in the closing stages.
Former Chelsea man Demba Ba forced a superb save from Petr Cech, but from the resulting corner Ersan Gulum pounced on a loose ball to bag the winner.
That meant Besiktas claimed the trophy following their 1-0 triumph over Fenerbahce in the evening's first contest.
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