He was elected with 87 votes out of 97 lawmakers who attended the session.
Barzani was nominated by the Kurdistan Democratic Party (KDP), which got the most seats in 2018’s elections, after it secured 45 seats in the 111-member assembly.
His election comes one day after his cousin Nechirvan Barzani was sworn in as president of the Kurdistan region.
The president will task him to form a new cabinet within 30 days.
The new prime minister-designate is the son of Masoud Barzani, the region’s first president, who held the post from 2005 until 2017.
In 2012, he became chancellor of the Kurdistan Region Security Council, in charge of the intelligence and other security services.
Kurdistan has an estimated population of 5.2 million people.