Alexandria Quigley, who once played for Sopron's Hungarian rivals MiZo Pecs, hit a three-pointer at the end of the third quarter for a 68-58 Kosice advantage.
The hosts stayed on top for the remainder of the contest, going in front by 15 before settling for an 11-point triumph.
Kosice shared the ball well, with their Serbian playmaker Miljana Bojovic particularly effective.
She finished the game with 16 points and a career-high 12 assists.
Norbert Székely's Sopron had won their opening game against Arras in Hungary so they dropped to 1-1 with the defeat.
Their guard, Shenise Johnson, is going to be a handful for opponents this year.
She ended up with 29 points, 11 rebounds and four assists.