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.