In a telephone conversation on Wednesday, the top Iranian diplomat also wished success for Barzani and the people of the Iraqi Kurdistan region.
The Iraqi Kurdish parliament on Tuesday elected Nechirvan Barzani, the nephew and son-in-law of veteran leader Masoud Barzani, as president of the autonomous region.
The deputy leader of the Kurdistan Democratic Party (KDP) won 68 votes from the 81 members present in the 111-seat chamber.
Last year, the KDP emerged as the largest bloc in the parliament following the September 30 election with 45 seats.
Nechirvan is the second person to hold the office of president. It was occupied by his uncle from its creation in 2005 until late 2017, when he quit following an independence referendum opposed by Baghdad.