The 23-year-old was deemed surplus to requirements at Stamford Bridge after disappointing loan spells with Fulham and Everton.
Nevertheless, newly promoted Burnley decided to take the chance and have signed Broja for a reported fee of £20 million.
Broja has put pen to paper on a five-year deal with the club, keeping him at Turf Moor until the summer of 2030.
Speaking to Burnley’s in-house media team, Broja said: “I am really excited to be here as a Burnley player.
“It’s a really positive time to be joining this club ahead of the start of the new Premier League season. I can’t wait to get going!”
He added: “I’m feeling good, I’m ready and I’m excited about the challenge of playing for this club.
“I can already feel the support from the Burnley fans – I can tell you all that I will be giving my all every time I pull on the famous Claret shirt.”