Thirty fighters are killed in the day-long battle according to the defence ministry.
They were also travelling in black GMC Suburban vehicles, the type used by US government convoys.
The attackers then stormed into the compound, threw stun grenades at a group inside a police office and shot dead an American soldier, the US military said.
They then fled with four US troops who had been attending a meeting to plan security measures for the 10-day Shia mourning ceremony of Ashura, which culminates on January 30.
|“The precision of the attack, the equipment used and the possible use of explosives to destroy the military vehicles in the compound suggests that the attack was well rehearsed prior to execution”
Lieutenant-Colonel Scott Bleichwehl, US military spokesman
“The precision of the attack, the equipment used and the possible use of explosives to destroy the military vehicles in the compound suggests that the attack was well rehearsed prior to execution,” Lieutenant-Colonel Scott Bleichwehl, the US military spokesman, said in the statement.
“The attackers went straight to where Americans were located in the provincial government facility, bypassing the Iraqi police in the compound.”
The military said it was “looking at all the evidence to determine who or what was responsible for the breakdown in security at the compound and the perpetration of the assault”.
Three days after the killings, the US military in Baghdad announced the arrest of four suspects over the attack and said they were detained after a tip from a Karbala resident. No further information was released about the suspects.