The Ivory Coast international joins the Premier League side on a five-year deal.
Cornet after joining the Clarets told the club's website: "I am excited to be joining Burnley in the Premier League and to play with and against many of the greatest players in the world.
"I have developed the greatest respect for Burnley and what the club has done. I welcome the challenge of hard work and am fully committed to doing what I can to contribute. I look forward to what we can achieve."
Burnley chairman Alan Pace said: "We’re delighted to have secured such a talented player with Champions League pedigree and we look forward to him joining up with Sean and the team.
“The wide position was identified as in need of strengthening and we saw Maxwel as the ideal player to fulfil this with talent, versatility, and pace.
"It’s been great to have the fans back at Turf Moor and I expect a brilliant reception for Maxwel when he is available for selection following the international break.”
In a seven-year stay with Lyon, Cornet scored 51 goals.
Cornet's last game for Lyon was in a 3-0 loss to Angers.
Cornet was on target in Lyon's Champions League quarter-final win over Manchester City in August 2020.
The 24-year-old scored 2 goals with five assists in thirty-six Ligue 1 appearances for Lyon last term.
