The 27-year-old has signed a four-year contract at Turf Moor after establishing himself as Brest’s first-choice goalkeeper last season.
"I'm really pleased to have joined Burnley and to be playing in England for the first time in my career,” he said.
"It's a great new chapter for me; I can't wait to get working with the team and I'm more than ready for this opportunity."
Born in Rodez, the Frenchman began his career at Tours FC, making his senior debut against Nancy in Ligue 2 in 2017.
After initially featuring for Brest’s reserve side, Coudert made his first-team debut during the 2023/24 campaign and gradually became a regular.
The goalkeeper has made more than 100 senior appearances in France and now moves to England to begin a new chapter with Burnley as they strengthen their squad.