A fee for the 20-year-old forward was not disclosed by the Bundesliga club but German media outlet Kicker report it to be €13 million.
Augustin has represented France at all youth levels from under-16 to under-21 but has found first-team opportunities at Parc des Princes limited, with nine of his 10 Ligue 1 appearances coming from the bench last season before the turn of the year.
"I am very happy to be a Leipzig player. I hope that I can help the team with my flexibility in attack," Augustin told a news conference.
Transfer chief Ralf Rangnick added: "We observed Jean-Kevin Augustin's development for a long time and are very happy that we reached an agreement."
Rangnick's main challenge during the close season might be keeping hold of stars such as Naby Keita – the midfielder who has been extensively linked with Liverpool – and playmaker Emil Forsberg as Leipzig look to build on last year's surprise second-place finish.
"We will definitely not be letting any key players go," Ragnick said. "There's nothing that will make us budge.
"It's totally normal for clubs to be interested in our players, but that doesn't affect our stance on the matter."