Behind them in the secret ballot was Egyptian career diplomat Moushira Khattab with 13 votes and China's Tang Qian with five, according to results posted on UNESCO's website.
Qatar's Hamad bin Abdulaziz al-Kawari and France's Audrey Azoulay are neck-and-neck in the race to head the UN's cultural body.
The victor must be approved by UNESCO's 195 member states in November, though this is seen as a formality. These two candidates had arrived ex aequo on Thursday night, with 18 votes each, behind Qatari candidate (22 votes) who has been racing in lead since beginning of selection process on Monday.
Arab countries have long wanted to lead UNESCO. In 2014, country had contributed $10 million to Unesco World Heritage Fund. His candidate, Hamad Al-Kawari, 69 years, was Minister of Culture and a familiar of United Nations, where he officiated as Vice-Chair of General Assembly.
Jewish groups have taken issue with Al-Kawari's candidacy, citing a preface the former Qatari Culture Minister wrote to a 2013 Arabic book called "Jerusalem in the Eyes of the Poets" that they accuse of anti-Semitism.
Khattab had come joint-second place with Audrey in the fourth round of voting on Thursday, hence the run-off to decide who would compete with Al-Kawari.
The ministry said the request made Friday was submitted by Egypt's permanent mission to UNESCO.
French media reported that Qatar recently invited several members of the UNESCO executive board on an all-expenses-paid trip to Doha.
UNESCO has been accused of bias in the Israeli-Palestinian conflict, and it infuriated Israel and staunch ally the United States by granting full membership to Palestine in 2011.
The next leader will have to allay fears over UNESCO's future as the agency faces a withdrawal from both the US and Israel.
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