The 22-year-old has signed a four-year deal, with the option for a further season, after leaving Elland Road, where he lost his starting spot to Kiko Casilla in January last season.
Peacock-Farrell spoke out about his situation while on international duty in the summer, saying he would look to leave Leeds if he was not playing regularly.
And though he is far from guaranteed regular action at Turf Moor given the presence of both Nick Pope and Joe Hart, the latter linked with a move himself, Peacock-Farrell said the opportunity was too good to turn down.
“I’m really happy to be here,” he said. “A Premier League club is what you dream of when you are starting out in football and to say I’ve signed for Burnley is a massive honour.
“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.
“So to come here and be part of it, is something I’m really excited about.”
Peacock-Farrell, who qualifies for Northern Ireland via an Enniskillen-born grandfather, came through the academy at Middlesbrough before joining Leeds in 2015.
He became first choice late in the 2017/18 season and began last year as the established pick, only to lose his place when Casilla joined from Real Madrid in January.
Now established as Northern Ireland’s first choice, Peacock-Farrell said he is looking forward to the next step in his career.
“I would say it’s probably a long-term project,” he added. “It was an opportunity I couldn’t pass up here.
“I’m not saying I’m coming here to replace anybody. I’m not. I’m here to learn and try and progress and be a better goalkeeper.
“I had my first session today and felt good. Hopefully I can hit the ground running and I’m looking forward to the weeks to come.”
Peacock-Farrell is Burnley’s third senior signing of the summer following the return of striker Jay Rodriguez, who has joined from West Brom, and the acquisition of defender Erik Pieters from Stoke.