The Revolutionary Guards is an armed force parallel to Iran's army which claims to defend the 1979 revolution against a possible coup.
The statement said the unit would not hesitate to confront "illegal" rallies organised by supporters of Mir Hossein Mousavi, the moderate politician who is contesting the re-election of Mahmoud Ahmadinejad.
"At the current sensitive situation ... the guards will firmly confront, in a revolutionary way, rioters and those who violate the law," the statement said.
The Iranian capital has seen unrest and street protests since results of the vote were announced on July 13.
Mousavi had renewed calls to his supporters to continue protests on Sunday.
The government is blaming the crisis on what it calls "terrorists" influenced by the West, and has said it will clamp down on any violent action.
The Revolutionary Guards has also warned Western countries against supporting the "rioters".
Over the weekend, clashes between police and anti-government protesters left at least 12 people dead and more than 100 wounded - raising the death toll to 19 since the unrest began.
Gunfire was heard in Tehran overnight, although state television reported the city calm on Monday.
Iranian state radio reported that more than 450 people had been arrested during Saturday's rallies, mostly around Tehran's Azadi square.
Forty police officers were also wounded, and 34 government buildings damaged, the Fars news agency reported.
Despite the deaths, arrests, and an earlier warning from Ayatollah Ali Khamenei, Iran's supreme leader, the demonstrators have appeared to be undeterred.
Alireza Zaker-Esfahani, an adviser to Mahmoud Ahmadinejad, the president, criticised Mousavi for not trying to calm his supporters.
"The weakness is in Mir Hossein Mousavi's political behaviour... He is currently issuing statements inviting his supporters to take to the streets. That will not solve any problem," he told Al Jazeera on Monday.
"Rallies will ultimately contribute to abuse, setting buses on fire, bloodshed and constant insecurity for the people."
He also said that once the security situation in the country has escalated, the president and other politicians have to step back and let security forces handle the situation.
"The security forces are the ones who should lay down plans and execute them, whereas Ahmadinejad, his interior ministry and all other political forces can only enter the scene if and when the security situation becomes one of political interactions. Ahmadinejad cannot do anything now," Zaker-Esfahani said.
Iran's Guardian Council, the country's highest legislative body, meanwhile, has admitted some irregularities occurred during the election.
Abbas-Ali Kadkhodaei, a spokesman for the council, told state-funded broadcaster IRIB on Monday that up to three million votes were under scrutiny, after it was found that the number of votes exceeded the number of eligible voters in 50 cities.
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However, he said it was a normal discrepancy because people are allowed to travel to other areas to vote, and that it was "yet to be determined whether the amount is decisive in the election results".
Ahmadinejad won the election by a wide margin, with 63 per cent of the vote, according to figures from Iran's interior ministry.
Mousavi received only 34 per cent of the vote, although he and his supporters allege voter fraud and have called for an annulment of the result.
In a statement published on the website of Mousavi's Kalameh newspaper on Sunday, the opposition leader said that Iranians had the right to protest against "lies and fraud", but also urged them to show restraint as they take to the streets.
"The revolution is your legacy. To protest against lies and fraud is your right. Be hopeful that you will get your right and do not allow others who want to provoke your anger ... to prevail," he said.
The Iranian government, meanwhile, has also cracked down on independent media reporting on the protests, and imposed severe restrictions on foreign journalists.
At least 23 journalists have so far been detained by authorities, according to the Reporters without Borders organisation, and a BBC correspondent has been expelled from the country.
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