With 90% of the votes counted, the pro-Rouhani List of Hope is set to take all 30 parliamentary seats in the capital.
The leading conservative candidate Gholamali Haddad-Adel is in 31st place.
Millions voted on Friday to elect the 290-seat parliament as well as members of the Assembly of Experts.
The 88-member assembly appoints Iran's Supreme Leader and might end up choosing a successor to Ayatollah Khamenei, who is 76 and has suffered ill-health.
Early results gave former President Akbar Hashemi Rafsanjani, a moderate conservative, and Mr Rouhani the most votes for the assembly, which is composed of mostly elder and senior clerics.
Analysis: Lyse Doucet, BBC News international correspondent
This stunning election result will make a difference in Iran's engagement with the wider world.
President Rouhani's hand has been strengthened in parliament to help open his country to greater trade and investment. That will help him, and others in his reformist camp, to deepen the dialogue with the West, which began with negotiations on a landmark nuclear deal.
But much of this opening will continue to be with Europe, rather than the US. Iran's relationship with America is still complex and controversial.
Iran's ambitions in the region are also deeply rooted - it has strategic interests in countries like Syria, Iraq and Lebanon as well as Afghanistan, and a strong sense of its right to remain engaged. These are areas where Iran's powerful Revolutionary Guards and its conservative Supreme leader hold sway.
But Iran wants to be regarded as an equal partner, able to sit at the world's top tables to work on common threats like the so-called Islamic State. President Rouhani's team may now feel empowered to engage a bit more, more often.
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