The latest deaths bring the number of people killed in the violence to at least 19, according to official sources.
A member of Iran's paramilitary Revolutionary Guard was also reported killed in the town of Najafabad.
Numerous videos showed some crowds of people chanting "Death to the dictator!" walking through the streets of Tehran.
Iran's "unity was a bullet (arrow) in their eyes and our progress and success in the world of politics and against the U.S. and the Zionist regime was not bearable to them", he said, adding that the Saudis "have blatantly said that we will create problems in Tehran". Some rioters tried to storm a police station to steal weapons.
The demonstrations - the largest to hit Iran since the protests around the disputed 2009 election - began in Mashhad, apparently over rising food prices, and expanded to several cities.
Speaking after a meeting with the heads of parliamentary committees on January 1, Rohani said the protests are not only related to economics but a cry from citizens for more freedom. Iran's Security Force is firing tear gas and the sound of gunshot is clearly heard.
Likewise, the Basij militia - which along with the IRGC led the crackdown against the 2009 protests that left dozens dead - has thus far appeared to stay away from the demonstrations.
That improvement has not reached the average Iranian, however.
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'I demand all prosecutors across the country to get involved and the approach should be strong, ' he said.
Ali Shamkhani, secretary of Iran's Supreme National Security Council, described the unrest as a "proxy war against the Iranian people".
US President Donald Trump has tweeted in support of the protesters, writing: "The great Iranian people have been repressed for many years". 'They are hungry for food & for freedom. "The guy who is totally against the Iranian nation has no right to sympathize with Iranians", Rouhani said.
The Russian Foreign Ministry said on January 1 that the unrest was "Iran's internal affair", adding that any "external interference destabilizing the situation is inadmissible".
He criticized the Iranian regime's response to the protests and also blamed European governments for watching "in silence" as the protests turn violent.
Nasserbakht said that 200 protesters were arrested on Saturday, 150 on Sunday and 100 were arrested Monday.
'We regret the loss of life that has occurred in the protests in Iran, and call on all concerned to refrain from violence and for worldwide obligations on human rights to be observed, ' he said. The country's foreign adventures were scorned as some demonstrators chanted, "Leave Syria, think about us!"
Israeli Prime Minister Benjamin Netanyahu praised the "brave Iranians" protesting against a regime that "wastes tens of billions of dollars spreading hate". "They have nothing to be happy about", Sarita Mohammadi, a 35-year-old teacher in Tehran, told Agence France-Presse.