Mousavi's statement appeared on the website of his Kalameh newspaper as reports emerged of police attacking a vigil by about 100 people outside the offices of the United Nations in Tehran.
The capital had been largely calm until then after state television reported that at least 13 people had been killed as police and pro-government militia clashed with protesters the previous day."I as one of the mourners invite my dear people to self-restraint. The nation belongs to you," Mousavi said.
"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."
Television reports on Sunday blamed rioters and "terrorist groups" for the previous day's violence, saying two petrol stations and a mosque had been set alight, and a military outpost attacked.
About 3,000 opposition protesters had tried on Saturday to enter the area, undaunted by a warning from Ayatollah Ali Khamenei, Iran's supreme leader, not to continue demonstrations against Ahmadinejad's re-election.
Security forces responded with live rounds, batons and tear gas, while protesters fought back with stones and set fires in the streets.
Mohsen Makhmalbaf, the foreign spokesman for Mousavi, defended the actions of the protesters.
"These people are in the streets to say 'We don't want atomic bombs, we want democracy'," he told Al Jazeera from Paris.
Witnesses said that dozens of people were hospitalised after being beaten by police and the pro-government Basiji militia.
The New-York based International Campaign for Human Rights in Iran said that scores of injured demonstrators had been arrested as they sought medical treatment.
It said doctors in Tehran's hospitals had been ordered to report injuries to the authorities, and that some seriously wounded protesters had sought refuge at foreign embassies in a bid to evade arrest.
"The arrest of citizens seeking care for wounds suffered at the hands of security forces when they attempted to exercise rights guaranteed under their own constitution and international law is deplorable," Hadi Ghaemi, the spokesman for the campaign, said.
Reports on community-driven websites such as Twitter claimed that a number of protesters were killed by police.
One video uploaded to YouTube on Saturday alleged to show a teenage girl - referred to as Neda on social networking sites - dying on the street after being shot by police.
As the clashes took place, local news agencies said that a suspected suicide bomber blew himself up outside the shrine of Ayatollah Ruhollah Khomeini, the leader of the Islamic revolution in 1979, injuring at least two people.
Government-run television said members of the exiled Mujahedeen-e-Khalq (MEK)opposition group were arrested in connection with Saturday's unrest.
The report claimed they were acting under British influence.
Al Jazeera was unable to verify the authenticity of the video or other reports of violence due to an official ban on independent reporting in the capital.
State broadcaster IRINN said 100 people were injured Saturday's violence.
Amnesty International (AI) cautioned that it was "perilously hard" to verify the casualty figures.
"The climate of fear has cast a shadow over the whole situation," Drewery Dyke, AI's chief Iran researcher, said.
"In the 10 years I've been following this country, I've never felt more at sea than I do now. It's just cut off."
Grand Ayatollah Hossein Ali Montazeri, who was key figure in the Islamic revolution in 1979 but is now under house arrest after falling out with the current religious leadership, criticised the crackdown and called for three days of mourning for the dead.
"Resisting people's demand is religiously prohibited," he said in a statement on his website.
Mohammad Khatami, a former president, also spoke out, criticising the decision to allow the Guardian Council, the country's top legislative body, to rule on the legitimacy of the vote.
|Five members of Rafsanjani's family were arrested after Saturday's protests [EPA]
"Referring the dispute to a body which has not been impartial regarding the vote, is not a solution," he said in a statement reported on the Mehr news agency.
The Guardian Council has offered a partial recount of ballots in order to appease protesters, but Makhmalbaf told Al Jazeera: "We want [a] revote, not [a] recount."
"We ask all people around the world not to confirm Ahmadinejad as our president," he said.
Five family members of Ayatollah Hashemi Rafsanjani, another prominent Iranian politician and former president, were reported to have been arrested on Saturday, including his eldest daughter, for taking part in an illegal protest.
Last week, state television showed images of Faezeh Hashemi speaking to hundreds of Mousavi supporters.
After her appearance, pro-Ahmadinejad students gathered outside the Tehran prosecutor's office and accused her of treason, state radio reported.
All apart from his daughter were later released.
Rafsanjani heads two very powerful groups in Iran: the Assembly of Experts, which can elect and dismiss the supreme leader, and the Expediency Council, a body that arbitrates disputes between parliament and the unelected Guardian Council.