Nick Pope – who missed much of last season through injury and then struggled to establish himself in the side ahead of Heaton and Joe Hart – is tipped to be Burnley's first-choice goalkeeper for the new season.
But extra competition will come from Peacock-Farrell, who has signed a four-year deal with an option for a further season.
The 22-year-old Northern Ireland international made his debut for Leeds in April 2016, but failed to become a regular at Elland Road until the second half of the 2017-18 campaign.
Peacock-Farrell was first choice for much of Marcelo Bielsa's first campaign in charge of the Championship club, though lost his place after Leeds signed Kiko Casilla from Real Madrid in January.
“I'm really happy to be here,” Peacock-Farrell told Burnley's official website. “A Premier League club is what you dream of when you are starting out in football.
“There's massive competition here, but it was an opportunity I couldn't turn down.
“The goalkeepers here are really good and two of them are England internationals. There’s something going on here that must be right.”
Doubts remain over Hart's future at Turf Moor. The former Manchester City and England star featured in Burnley’s first 19 league games last term, but failed to make a single appearance after a 5-1 defeat to Everton on Boxing Day.
Heaton made exactly 200 appearances for the Clarets before departing for Villa Park yesterday, and didn’t miss a single league fixture for a total of three seasons between 2013 and 2016.
Pope then took the gloves for the majority of the 2017-18 season, but suffered a dislocated shoulder in a Europa League clash with Aberdeen last summer – an injury that omitted him from team selection until the turn of the year.
The club, however, rewarded the 27-year-old with a new four-year deal in May 2019, making him favourite to hold down a place in the starting XI this season.