There was no information of casualties in Saturday's offensive.
Local police were warned to stay inside and unarmed during the operation, 120km south of Baghdad.
The raids come ahead of demonsrations by supporters of al-Sadr planned for Monday to mark four years since the fall of Baghdad.
The US military said air support was called in by the Iraqi army following the "positive identification of ... illegally armed militia men using shoulder-fired rocket-propelled grenades".
Battle for Baghdad
The Iraqi military source said clashes were pitting Iraqi and US forces against armed men in Diwaniyah's Salim Street and southern areas of the city including the Nahda and Wahda districts.
One civilian is reported to have been killed and five others wounded in the fighting.
A curfew continued in Diwaniyah for a second day and hospitals were calling in doctors to report for duty, the Iraqi army source added.
US-backed Iraqi forces swept through the city on Friday, arresting nearly 30 people and killing three fighters as part of the crackdown aimed at Shia groups.
The military said the operation was intended to disrupt militia activity and return control of the city south of Baghdad to the government.
Over the past year, Diwaniyah has seen repeated clashes between the security forces and Shia militias.
Last August, nearly two dozen Iraqi soldiers and dozens of militiamen were killed in the deadliest bout of fighting.
Güncelleme Tarihi: 20 Eylül 2018, 18:16