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.
Manuchar Markoishvili, BC Kyiv player:
"We have a chance to win at home, but we have to be more concentrated and better in offensive rebounds. We have to increase our level [so we can win] and keep hope to go to a third game."
Antywane Robinson, Cholet player:
"My knee is better every day and I am fine. Yesterday [coach] Erman [Kunter] asked me to prepare for this game. I was feeling great before the game. It is a good victory for the team. We have made a big step towards the Final four. Now we have to win this very important next game in Kiev."
Erman Kunter, Cholet coach:
"This victory is good for everybody. It is only one game, we can be satisfied but we have to continue in this way. Tonight the public made the difference, they were great supporting the team and gave them energy in defense. It was a hard game. The philosophy of this team is to prevent their opponent to play their game. For the next game everything is possible. But we shouldn't give them a chance. We have to win in Kiev."