The 19-year-old has signed a five-year deal, which expires in 2017, and becomes the club's third signing of the summer window.
“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 made his debut last season at the age of 18, after Matteo Sereni cancelled his contract with Brescia following their relegation.
The young shot-stopper went on to make 17 appearances in all competitions.