Barzani, the Kurdistan Democratic Party (KDP) candidate, got 68 votes, while all four of his rivals got none.
The Patriotic Union of Kurdistan (PUK), with its 21 deputies, issued a statement ahead of the session saying the KDP has not respected prior agreements. They therefore did not participate in the vote.
The session saw the participation of 81 of the 111-strong parliament.
Nechirvan Barzani, nephew of former president Masoud Barzani, will have to step down as prime minister to assume the role of president.
He is expected to ask his cousin, Masrour Barzani to form the new cabinet, in June.