The reformist alliance formed in support of moderate President Hassan Rouhani had campaigned against both Yazdi, the outgoing chairman of the Assembly of Experts, and Mesbah-Yazdi, an outspoken opponent of the reformists.
But a third conservative, Ahmad Jannati, who had been targeted by the pro-Rouhani coalition, narrowly kept his seat on the assembly, a powerful committee which oversees the work of supreme leader Ayatollah Ali Khamenei, state television reported.
Jannati, who chairs the conservative-dominated Guardian Council which must confirm the results of Friday's elections for both the assembly and for parliament, came 16th in Tehran, scraping the last of the capital's 16 assembly seats.
Verification of the result is not expected for several days.
The election to the 88-member Assembly of Experts is important because it will pick Khamenei's successor if he dies during its eight-year term.
Rouhani placed third in polling for the assembly in Tehran and his key ally, former president Ayatollah Akbar Hashemi Rafsanjani took first place.
Signup to our newsletter and follow us on Facebook and Twitter!
The race for the Assembly of Experts saw claims from prominent conservatives of foreign interference and some accused the Rouhani-Rafsanjani slate of being "a British list".
On Sunday as it became increasingly clear that Yazdi and Mesbah-Yazdi were set to lose their seats the head of the country's judiciary, Ayatollah Sadegh Larijani echoed those allegations.
Reformists worked with "American and English media outlets" during the poll, Larijani charged.
"Is this type of coordination with foreigners in order to push out these figures from the Assembly of Experts in the interests of the regime?" he asked.
Rouhani's government dismissed the accusations when they were first made during the election campaign.
"We don't have anything such as a British list and if anyone wants to say that there is such a list, they are in fact insulting the Guardian Council," said government spokesman Mohammad Bagher Nobakht, a vice president.
The Guardian Council approved all candidates who ran in the elections for parliament and for the Assembly of Experts.