The escalating tensions between the two adversaries may imperil efforts to end the wars gripping Syria and Yemen, where Saudi Arabia and Iran back rival sides.
Saudi Arabia says last week's executions were a domestic matter, and that Iran is the country pursuing sectarian division by casting itself as the champion of Arab Shi'ites.
They said an air strike had hit a public square about 700 meters (yards) away from the embassy and that some stones and shrapnel had landed in the embassy's yard.
Asseri acknowledged that coalition jets carried out heavy airstrikes in Sana'a on Wednesday night, targeting missile launchers used by the Houthi militia.
A spokesman from the Saudi-led coalition, Brigadier General Ahmed Asseri, said the claims were investigated.
Shortly after the unrest, a diplomatic row erupted in the Middle East, with Saudi Arabia severing diplomatic ties with the region's main Shiite power, Iran.
The Times newspaper reported on Tuesday that British Prime Minister David Cameron had delayed a trip to Saudi Arabia.
A Saudi-led coalition has been bombing Yemen since March, in an attempt to repel the Shia rebel Houthi movement, which is allied with Iran.
Iran's official state broadcaster IRIB said the diplomats had arrived in Tehran, showing pictures of their plane after it landed at Tehran's Mehrabad Airport.
Saudi Arabia cut diplomatic relations with Iran late on Sunday, hours after protesters stormed and set fire to the Saudi Embassy in Tehran in protest against the execution.
Speaking to reporters on Thursday, Amir Abdollahian pointed to a recent rocket attack by the Saudi regime near the Iranian embassy in Sana'a, saying that unfortunately, one of the embassy guards has been severely injured.
Jordan and Kuwait have both summoned their envoys to Iran and the United Arab Emirates has partially downgraded its diplomatic relations with Tehran. It said the decision came during an emergency meeting of the Cabinet of President Hassan Rouhani.