The 27-year-old striker has signed a three-year deal and Burnley’s seventh summer signing.
“I have no doubt this is an ideal move for me,” he told burnleyfootballclub.com.
“Now the opportunity is here and I’m very thankful to the football club to give it to me. I feel I can go on and do great things.”
Wells is returning to full fitness following minor ankle surgery earlier this summer.
Burnley’s next Premier League match is on Sunday 10 September at home to Crystal Palace.