The 19-year-old has signed a five-year deal with the Italian champions and is reported to have cost Juventus 3.8 million euros.
"It’s a dream come true,” Leali told the club's official website. “Joining such a prestigious club at my age is extremely gratifying and a great motivation to keep developing further.
"Being part of the same club as a champion like Gigi Buffon is hugely satisfying, but it also makes me realise how much I’ll have to work for the opportunity to repay the trust placed in me.”
Leali is a youth international for Italy and last season made 16 appearances for Brescia in Serie B.