Iranian protesters stormed the Saudi Embassy in Tehran early yesterday and Shia Muslim Iran's top leader predicted "divine vengeance" for Saudi Arabia's execution of a prominent Shia cleric.

Demonstrators protesting against cleric Nimr al-Nimr's execution broke into the embassy building, smashed furniture and started fires before being ejected by police.

Iran's President Hassan Rouhani condemned the execution as "inhuman", but also called for prosecuting "extremist individuals" for attacking the embassy and the Saudi consulate in the northeastern city of Mashhad, state media reported.

Tehran's police chief said an unspecified number of "unruly elements" were arrested for attacking the embassy with petrol bombs and rocks. A prosecutor said 40 people were held.

Iranian Supreme Leader Ayatollah Ali Khamenei, criticising Saudi Arabia for the second straight day over Nimr's execution, said politicians in the Sunni kingdom would face divine retribution for his death, reports Reuters.

"The unjustly spilled blood of this oppressed martyr will no doubt soon show its effect and divine vengeance will befall Saudi politicians," state TV reported Khamenei as saying.

UN chief Ban Ki-moon said he was "deeply dismayed" by Saudi Arabia's execution of 47 people including the prominent Shia cleric, reports AFP.

Deploring the violence outside the Saudi embassy in Tehran, he called for "calm and restraint" and urged "all regional leaders to work to avoid the exacerbation of sectarian tensions," his spokesman said in a statement.

The US State Department said Nimr's execution "risks exacerbating sectarian tensions at a time when they urgently need to be reduced". The sentiment was echoed almost verbatim by EU foreign policy chief Federica Mogherini and an official at the German Foreign Ministry. The State Department also urged the Saudi government to "respect and protect human rights”.

UN High Commissioner for Human Rights Zeid Raad al-Hussein said it was not clear those killed were granted effective legal defence, while the scale of the executions was very disturbing "particularly as some of those sentenced to death were accused of non-violent crimes".

Iran's Revolutionary Guards had promised "harsh revenge" against the Saudi Sunni royal dynasty for execution of the Shia cleric.

Nimr, the most vocal critic of the dynasty among the Shia minority, had come to be seen as a leader of the sect's younger activists, who had tired of the failure of older, more measured leaders to achieve equality with Sunnis.

Although most of the 47 men killed in the kingdom's biggest mass execution for decades were Sunnis convicted of al-Qaeda attacks in Saudi Arabia a decade ago, it was Nimr and three other Shia, all accused of involvement in shooting police, who attracted most attention in the region and beyond.

Human rights groups say the kingdom's judicial process is unfair, pointing to accusations that confessions have been secured under torture and that defendants in court have been denied access to lawyers. Riyadh denies torture and says its judiciary is independent.

Analysts have speculated that the execution of the four Shias was partly to demonstrate to Saudi Arabia's majority Sunni Muslims that the government did not differentiate between political violence committed by members of the two sects.