Ayatollah Yazdi, 83, was a deputy speaker of parliament after the 1979 Islamic Revolution and headed the judiciary for a decade until 1999.
He gained 47 of the 73 votes cast at a closed-door meeting in Tehran, according to the website of national television, citing officials.
Ayatollah Yazdi was among five contenders whose names had been linked to the post by Iranian media in recent weeks.
Another prominent cleric, Ayatollah Mahmoud Shahroudi, withdrew before the vote.
The election comes at a sensitive time as talks over a deal on Iran's nuclear program reach a critical stage.
His election represents a heavy defeat for former president Ayatollah Akbar Hashemi Rafsanjani, who previously held the position between 2007 and 2011, and who received 24 votes.
Ayatollah Yazdi takes up a position vacant since October 2014, when Ayatollah Muhammad Reza Mahdavi Kani passed away following a heart attack.
Officially comprised of 86 religious figures elected by the people, the Assembly of Experts chooses the Leader and monitors his actions.
The clerical body grants the leader, currently Ayatollah Seyyed Ali Khamenei, an indefinite term but it retains the power to sack him if it sees fit.
Ayatollah Yazdi, initially at least, will hold the post for just one year as elections for the Assembly of Experts are scheduled alongside parliamentary polls next year, with a new vote for chairman to follow.
Although Rafsanjani put his name forward, he had said his membership of Iran's top political arbitration body, the Expediency Council, already kept him busy.
"Psychologically, I am not at all prepared to become the chairman of the Assembly of Experts. I work enough at the Expediency Council and why would it be necessary to work more than this?" he said in an interview published by the Shargh newspaper on Tuesday.
"We'll see on the day. My criterion is that it should be someone who befits the stature of the Assembly of Experts."
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