Pierre Vincent's club, fighting for their lives in the competition after losing three straight games, improved to 3-4 with the victory - their second against Kosice this season.
The triumph lifted the French club to third in the standings above Kosice, who are also 3-4.
Essence Carson had 22 points, five rebounds and four assists in the Bourges win.
Bourges led just 13-11 when they finished the first quarter with an 8-3 run.
Vincent's side stayed in front for the remainder of the contest.
Kosice did match the French blow for blow in the second quarter and only trailed 42-35 at half-time.
In the third quarter, however, Vincent's team pressed home the advantage by outscoring the visitors 22-15.
Bourges got a big boost from their Greece international Stella Kaltsidou, who had a season-high 18 points.
Angel McCoughtry scored 20 points and grabbed seven rebounds for Kosice, whose Candice Dupree contributed 14 points and nine boards.