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.
"Everybody knows that Bourges is not in the list of the top-five most offensive teams. The most important thing is to find good positions [to take a shot]. Once you do so, then it depends on whether you are on a scoring day or not. I'm here to find solutions so even if we couldn't score in the beginning of the match I was never afraid. Tonight I told the girls to make sure there are no easy shots for Mc Coughtry and we succeeded in that. Kosice defended in zone, individually and sometimes match-up but we found a solution every time."
Stefan Svitek, coach Good Angels Kosice:
"Tonight we had to fight against a very good team of Bourges, who played strong defense. It was so hard to score. In the beginning both team was really nervous, but we were positively aggressive. But then we lost that and and we were not able to stop them. They scored more than 20 points in every quarter. I think our victory against Gorzow could prove a good thing for us but we will see."