Rodrigue Beaubois, the exciting 21-year-old Guadeloupe-born guard, scored 18 points to lead the French team.
Sixteen of those points came in the second half when Cholet put the result beyond doubt.
Beaubois has now scored 18 in his last three games.
Nando De Colo weighed in with 16 points for Cholet.
BC Kyiv's Ukraine international Artur Drozdov hit a three-pointer to cut the Cholet lead to 31-30 early in the third quarter, only for Beaubois to answer with a trey to push the lead back to four.
After Denys Lukashov again cut the deficit to one with a shot from the arc, Beaubois struck again from long range.
Randal Falker then made two free throws, and both Beaubois and Vincent Grier went in for lay-ups to open up a 43-33 advantage.
Kyiv got no closer than seven points the rest of the way.
Drozdov scored 13 points to lead Kyiv.
Also for Cholet, Randal Falker had 11 rebounds to go with his four points and Claude Marquis eight points with 10 boards.
Game 2 is on Thursday in Ukraine.