In Mashhad, Iran's second biggest city, demonstrators meanwhile set fire to the Saudi consulate, according to news sites, carrying pictures of the alleged assault.
The incidents came hours after the announcement of the death of 56-year-old cleric Nimr al-Nimr, a key figure in anti-government protests in the kingdom since 2011.
The execution prompted strong condemnation from Shiite-majority Iran and Iraq.
"There are flames inside the embassy... demonstrators were able to get inside but have since been cleared out," ISNA said.
"The fire has destroyed the interior of the embassy," an eyewitness told AFP. "The police are everywhere and have dispersed the demonstrators, some of whom have been arrested."
Protesters had been able to climb up onto the roof of the embassy before they were made to leave, ISNA added.
Websites carried pictures of demonstrators apparently clutching the Saudi flag, which had been pulled down.
Iranian media quoted foreign ministry spokesman Hossein Jaber Ansari as asking police to "protect Saudi Arabia's diplomatic missions in Tehran and Mashhad... and prevent any demonstrations in front of these sites."
Nimr, who spent more than a decade studying theology in Iran, was among a group of 47 Shiites and Sunnis executed Saturday on charges of terrorism.
Predominantly-Shiite Iran, the Sunni kingdom's longtime rival, said in reaction to Nimr's execution that "the Saudi government supports terrorist movements and extremists, but confronts domestic critics with oppression and execution."
It will "pay a high price for following these policies," Jaber Ansari had warned before the attacks took place.
In response, Saudi Arabia's foreign ministry said it had summoned Iran's envoy to protest at the "aggressive Iranian statements on the legal sentences carried out today".
The Saudi interior ministry said the men had been convicted of adopting the radical "takfiri" ideology, joining "terrorist organisations" and implementing various "criminal plots".
An official list published included Sunnis convicted of involvement in Al-Qaeda attacks that killed Saudis and foreigners in 2003 and 2004.