The death of the Revolutionary Guard is said to be the first fatality among Iranian security forces during the protests.
The Iranian foreign ministry had also on Saturday strongly deplored the recent comments by Trump and State Department Spokeswoman Heather Nauert in support of the recent economic protests in Iran, describing them as "cheap, worthless and invalid".
Earlier on Monday, it was reported that the number of civilians killed amid the pro-and anti-government demonstrations has grown to 10.
Numerous videos showed some crowds of people chanting "Death to the dictator!" walking through the streets of Tehran.
A social media image showing protests in Tehran on December 30. Others have targeted the country's religious establishment.
The clashes began in Mashhad on Thursday before spreading to several other cities.
The unrest "may seem to be a threat, but it can be turned into an opportunity to see what the problem is", the Fars news agency quoted him as saying.
"Those who damage public property, disrupt order and break the law must be responsible for their behaviour and pay the price", interior minister Abdolrahman Rahmani Fazli said on Sunday. More than 400 people have been arrested in the protests, AFP said.
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. Unemployment in the spring of 2017 was 12.6 percent, up slightly from late 2016, according to the World Bank.
In a statement on Twitter, the ministry cautioned that Iran was seeing "large-scale disturbances and an unstable security situation" as well as "grave acts of violence".
The Iranian government shut down access to social media platform Instagram on Sunday, as well as a popular messaging app, which protesters used to organize. "It's unclear how this is happening from town to town, except through social media".
"Hashtags and messages about the situation in Iran come from the United States, Britain, and Saudi Arabia", he said on January 1.
US President Donald Trump, who has been tweeting in support of the protesters, continued into the New Year, describing Iran as 'failing at every level despite the bad deal made with them by the Obama Administration'.
Nine people were killed overnight in unrest related to protests in the Isfahan region of central Iran, according to the state broadcaster. "They're also demanding more freedoms", said Rowhani, in an indirect attack on regime hardliners who oppose his attempts to push through political and cultural reforms.
He also warned that "the US is watching for human rights violations".
"The people are absolutely free in expressing their criticisms and even protests", Mr Rouhani told his cabinet in remarks reported on state media.
Elsewhere, British Foreign Secretary Boris Johnson has called for Iran to engage in "meaningful debate" about the "legitimate and important" issues raised by the protesters.
He also called on "all concerned to refrain from violence and for worldwide obligations on human rights to be observed".