Thursday's fighting took place in Kut, 160km southeast of Baghdad, a local police officer said.
He said eight militia members were killed; the US military said in an e-mail to The Associated Press news agency that it killed an "estimated" 11 fighters.
In a later statement, the US military said the operation was targeting a suspect who was "reportedly responsible for attacks against coalition forces".
Meanwhile, a proposed amnesty bill for a portion of prisoners being held in Iraq's prisons has come under criticism from a Sunni parliamentarian.
Iraq's cabinet on Wednesday approved a draft of a general amnesty bill for detainees being held in Iraqi prisons, a measure that could go a long way towards reconciling Iraq's antagonistic sects and factions.
Asmaa Adnan al-Dulaimi of the Iraq Accordance Front, the three-party Sunni alliance that has 44 parliamentary seats, told The Associated Press that the law "will hinder the release of many innocents" as it will encounter endless debate in parliament.
"The best thing is to leave this issue to the judicial system because it is the only side who can decide who is innocent and who is not," she said.
"The judicial system should review the inmates' files carefully and immediately in order to have them freed and not stranded by the long political discussions."
Not until March
In any event, the measure will not be brought to parliament for debate until March at the earliest, according to Sami al-Askari, a key adviser to Nuri al-Maliki, the Iraqi prime minister.
Al-Askari further said the amnesty would not cover those convicted of terrorism, corruption, crimes against humanity and kidnapping.
Many key draft laws - including measures to share oil revenue and to allow some members of Saddam Hussein's Baath party to hold government jobs - have remained mired for months in Iraq's paralysed parliament.