The former Borussia Dortmund coach has signed a deal until 2028, replacing Ernesto Valverde with the Bilbao sitting eighth in the Spanish league standings.
Terzic, 43, arrives in the Basque Country with a reputation for leading teams who play high-intensity, attacking football.
His tenure at Dortmund was marked by a German Cup title in 2021 and a run to the Champions League final in 2024, which they lost 2-0 to Real Madrid.
The German has been without a job since leaving Dortmund at the end of the 2023-24 season.