The Serbia international flew in early this morning to Turin and had all his various tests before putting pen to paper.
The Lega Serie A website, which has a register of all the contracts deposited with them, confirmed Kostic is now a Juventus player.
West Ham United had also tried to purchase the 29-year-old, but he always had the Bianconeri as his preference.
Because the move was imminent, Kostic did not participate in last night’s UEFA Super Cup, which saw Eintracht beaten 2-0 by Real Madrid in Helsinki.
He helped them win the Europa League last season, contributing seven goals and 15 assists in 43 competitive matches.
Meanwhile, Juventus have agreed to loan full-back Luca Pellegrini to Eintracht Frankfurt and he is expected to have his medical tomorrow.