The centre-back was strongly linked with a move to Manchester United before joining Salzburg from Valenciennes in 2015, off the back of helping France triumph at the Under-17 European Championships.
Barcelona, Bayern Munich, Arsenal and Juventus are all believed to have tracked the teenager in recent months, but Leipzig, who are owned by the same company as Salzburg, have won the race.
"We are delighted that Dayot decided to join us to make the next step in his career, despite several offers from top international clubs," Leipzig sporting director Ralf Rangnick told the club's official website.
"We are convinced that Dayot can already be a reinforcement for the first-team squad in the near future. Dayot has shown a great deal of promise for his age and we think he will fit in perfectly with the philosophy of the club."
Upamecano has signed a contract with the Bundesliga outfit until the summer of 2021.