The results, carried on state news agency IRNA, suggest conservatives may lose their dominance of the 88-member body, which is tasked with choosing the next supreme leader, the country's most powerful position, as reported by Reuters.
The Friday elections for the assembly and the parliament were the first since Rouhani signed a deal with world powers last July to limit Iran's nuclear activities in return for an easing of economic sanctions.
The twin polls were seen by analysts as a potential turning point for Iran and a vote of confidence in Rouhani's government and his detente policy with the West.
Rouhani's allies were also due to take all 30 parliamentary seats in the Tehran constituency, according to preliminary results. But their gains outside the capital were more limited, with conservatives keeping hold of many seats in both bodies.
The hardline chairman of the Assembly of Experts, Mohammad Yazdi, lost his seat. So did Mohammad-Taghi Mesbah-Yazdi, an arch-conservative who was widely seen as the spiritual mentor to former conservative president Mahmoud Ahmadinejad.
The only hardliner to make the cut was Ahmad Jannati, who squeezed in at 16th place. Jannati is also the chairman of the Guardian Council, a cleric vetting body that disqualified the majority of reformist candidates from running for the elections.