Pusic, who led Shakhtar Donetsk to the Ukrainian league title in 2024, has signed a contract until 2028.
The 54-year-old managed UAE side Al Jazira last season in his first coaching spell in the Gulf region.
He helped FC Twente secure promotion to the Eredivisie in 2019.
Pusic won the Eredivisie title as Feyenoord's assistant coach in 2023.
Jaissle, who guided the Saudi side to back-to-back AFC Champions League Elite titles, was named Newcastle United manager following the departure of Eddie Howe.