In separate but coordinated attacks the armed men, driving in three cars, opened fire on the buildings with automatic weapons and rocket-propelled grenades (RPGs), said police officer Issam Yasin.
In the first incident, up to 50 attackers stormed the Iraqi Civil Defence Corps (ICDC) compound, firing mortars, explosives and light machine guns. The assault sparked a fierce exchange of gunfire with ICDC members.
In earlier reports, Aljazeera's correspondent said 19 people were killed in that attack. "Their weapons were more powerful than our Kalashnikovs," a police officer was quoted as saying.
Fourteen of the slain were police officers.
Two hospitals in Falluja, revising a toll of 22 dead, said on Sunday the raid killed 22 police, one civilian and four attackers.
The attackers went from room to room, freeing about 100 prisoners held in the compound, another police officer said, speaking on condition of anonymity.
It was not known if the escapees included suspected resistance fighters.
The armed men also attacked the local mayor's office, about a kilometre away. There, two Iraqi policemen were killed, according to Aljazeera's correspondent.
A hospital official confirmed several people had been brought in for treatment.
Plain-clothed ICDC members
secure their base
Of the wounded, most were policemen, although two women and a child were among the injured, a security official at Falluja hospital, Abd al-Hamid al-Janabi said.
Two wounded fighters brought to the hospital were arrested, he added.
Police were still on high alert hours after the attack, maintaining positions on the rooftops of police stations and government buildings.
Aljazeera's correspondent reported that the situation in the town was tense after a curfew imposed to "control security".
Fighters in Falluja, a hotbed of resistance activity 50km west of Baghdad, have frequently attacked US troops and Iraqis who cooperate with them, especially police.
The attack came two days after fighters carried out a bold attack against the same ICDC base, just as the top US commander in the Middle East, General John Abizaid, was visiting.
In that attack, fighters opened fire with RPGs and automatic weapons from rooftops after Abizaid's convoy pulled into the walled compound.
Abizaid escaped unharmed and there were no US troop injuries. Police reported two Iraqis were killed.