Braswell poured in 26 points as Cholet prevailed 85-68 to remain undefeated in EuroChallenge Top 16 Group I.
The visitors from Holland slipped to 1-2.
Cholet won with a big fourth quarter.
The home side only led 56-55 with 8:30 remaining in the game when Braswell took over.
The 30-year-old American guard scored nine straight points with two jump shots, a lay-up and then a three-pointer for a 65-55 lead.
Thomas Larrouquis (18 points) then capped the run with another three-pointer 5:09 from the end.
Cholet pulled way over the last half of the quarter for a decisive win.
Claude Marquis finished with 19 points and nine rebounds for Cholet.
Anthony Richardson and Dean Oliver scored 16 points apiece for the visitors.
Erman Kunter (Coach Cholet Basket):
"Tonight it was a game like we could often see in EuroChallenge. Both teams were playing strong until the third quarter. At the beginning, EiffelTowers was driving the game and playing with high intensity, but in the second period our team took control of this game. Our two next games against Liège and Bonn will be at home, but it will not be easy to win them. Now we had to be focused on the game against Chalon in the French League on Friday."
Steeve Ho You Fat (Cholet Basket):
"It is difficult for a young player to come on the court in this kind of game, but when you begin to play, you forget your fears and try to do your best. EiffelTowers is a good team. They play hard and it was difficult for us to take control of this game. Kevin Braswell and Thomas Larrouquis made some good shots in the last quarter to help us finally win the game."
Don Beck (Coach EiffelTowers):
"Cholet played extremely well. I congratulate them. They are very athletic, very deep and good on the outside shots. We played not as athletic as them. We had a bad third and fourth quarter but we took a lot of good shots and I'm very proud of my team."
Anthony Richardson (EiffelTowers):
"We played a good game at the beginning, but in the third and fourth quarter, Cholet played great and we didn't succeed to stop them raising [their game]."