The tomb of Salman Pak, also known as Salman al-Farisi, was attacked after sunset with two rockets, said Jamal al-Saghir, an aide to Shia political leader Abd al-Aziz al-Hakim.
The tomb is located in the village of Salman Pak, 30km southeast of Baghdad. The village carries the name of the man.
The attack comes two days after a Shia shrine in the central city of Samarra was heavily damaged by an explosion.
Baghdad, meanwhile, stayed relatively calm on Friday, despite reports of minor clashes between members of a Shia militia and armed men in the south of the city.
Police sources said Friday's clashes were between unidentified gunmen, possibly Sunnis, and members of the Jaish al-Mahdi (the al-Mahdi Army) loyal to Shia figure Muqtada al-Sadr.
Al-Sadr had called on his followers not to attack Sunnis or their mosques.
It was not immediately clear if anyone was hurt in the skirmishes in the Saidiya area, which undermined a day-time curfew announced by the government to stop the spread of violence.
Residents, who are both Sunni and Shia, said they heard sustained gunfire overnight.
Iraqis are staying off the streets in the capital and surrounding provinces, where tensions have been running high since
Wednesday's attack on the Shia shrine sparked reprisals against Sunni mosques.
But many thousands of Shia flouted the curfew to fulfil their religious obligation and go to the mosques on Friday in the sprawling Shia slum of Sadr City, in the east of the capital, where support for al-Sadr runs high.
His al-Mahdi Army has been seen roaming the streets over the past two days, and Sunnis accuse it of being at the forefront of sectarian violence that has shaken Iraq's unity.
A curfew was imposed in parts of
Iraq after the mosque attack
Twenty bodies of people who were killed overnight and this morning have been brought to the mortuary, a police source said.
About 200 Iraqis have been killed in Baghdad alone since Wednesday, prompting officials and politicians to appeal for calm in an atmosphere of growing concerns of a rise in violence.
Attacks in Iraq
In Basra, where the curfew was not in effect, armed men on Friday kidnapped three children of a Shia legislator. The son and two daughters of Qasim Attiya al-Jburi - aged between seven and 11 years - were abducted by armed men near the family home, police said.
Al-Jburi is a member of the Islamic Dawa Party-Iraq Organisation and is the former head of Basra's provincial council.
Elsewhere, police found the bodies of two bodyguards for the Basra head of the Sunni Endowment, a government body that cares for Sunni mosques and shrines. They had been shot.
"This is the first time that I have heard politicians say they are worried about the outbreak of civil war"
Late on Thursday, Iraqi state television announced an extension of the night-time curfew until 4pm on Friday in Baghdad and the nearby provinces of Diyala, Babil and Salah al-Din, where the shrine bombing took place.
Gunmen stormed a house and killed two Shia men and a woman in Latifiya, southwest of Baghdad, at about 7am on Friday, despite a curfew. Two children, aged about 11 and 13, were wounded in the attack.
Police Captain Ibrahim Abd Allah, said five men,too, were killed in the attack.
Aim of curfew
The curfew was aimed at preventing people from attending the week's most important Muslim prayer service, which officials feared could be both a target for attacks and a venue for stirring sectarian feelings.
Such sweeping daytime restrictions indicated the depth of fear within the government that the crisis could set off a Sunni-Shia war.
Mahud Uthman, a Kurdish elder statesman, told The Associated Press that: "This is the first time that I have heard politicians say they are worried about the outbreak of civil war."