The victors were led by French power forward Adrien Moerman who scored 20 points at Istanbul's Sinan Erdem Sports Hall. Moerman also had seven boards.
Serbian point guard Vasilije Micic added 19 for Anadolu Efes.
French power forward-center Mouhammadou Jaiteh was the game's top scorer with 26 points for Empera Hali Gaziantep Basketbol, to go along with seven rebounds.
Game 2 of the best-of-three series will be held on Empera Halı Gaziantep Basketbol's home court Sunday, May 16 with Anadolu Efes looking to book a ticket into the semifinals.
Quarterfinals game 1 results:
Pinar Karsiyaka - Turk Telekom: 82-79 (1-0)
Fenerbahce Beko - Darussafaka Tekfen: 103-91 (1-0)
Anadolu Efes - Empera Hali Gaziantep Basketbol: 96-73 (1-0)
TOFAS - Besiktas Icrypex: 90-92 (0-1)
Game two fixtures
Saturday:
Turk Telekom - Pinar Karsiyaka
Darussafaka Tekfen - Fenerbahce Beko
Sunday, May 16:
Empera Hali Gaziantep Basketbol - Anadolu Efes
Besiktas Icrypex - TOFAS