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State media quoted a foreign ministry spokesman as saying planes had deliberately targeted the site.
But some later reports in Iran said missiles had struck only in the vicinity of the embassy.
Residents and witnesses in Sanaa reported there was no damage to the main embassy building.
Although the incident may turn out to be less serious than initially feared, the BBC’s Arab Affairs Editor Sebastian Usher says the growing row between Saudi Arabia and Iran could derail peace efforts in Syria and Iraq, as well as in Yemen.
The row began after Sunni-ruled Saudi Arabia executed a Shia cleric, Nimr al-Nimr.
Iranian protesters in Tehran, angry at the execution, then attacked the Saudi embassy, leading Saudi Arabia to cut off diplomatic relations.
A number of Saudi allies have followed suit in taking diplomatic action against Iran.
Meanwhile on Thursday Iran banned the import of all Saudi goods, according to a statement read on Iranian state TV.
Analysis: BBC security correspondent Frank Gardner
Ever since the Saudi-led air campaign began against Yemen’s Houthi rebels last March, there was always a risk that the cold war between the region’s two big rivals, Saudi Arabia and Iran, could ignite into something more serious.
The Saudis accuse Iran of smuggling in arms by sea to equip the Shia Houthis, who retain control over the capital and much of the country. Saudi officials have even claimed that Iranian military commanders are on the ground there, helping to direct the Houthis.
Both Iran and the Houthis deny this. The reality is that the Houthis owe most of their military gains to support from renegade Yemeni army units loyal to ousted ex-President Ali Abdullah Saleh.
Saudi Arabia’s military spokesman says its coalition has asked for precise co-ordinates of foreign embassies in the Yemeni capital so it can avoid hitting them. Angry as the Saudis are about the ransacking of their embassy in Tehran, it would have been a major escalation if they had carried out a deliberate, direct hit on Iran’s embassy in retaliation.
The Saudis accuse Iran of supporting the Houthis in Yemen militarily – a charge it denies.
It is not clear whether the Iranian embassy was fully functioning, but a number of countries have withdrawn their staff or relocated missions to the port city of Aden in the past year.