The violence erupted at about 3:00 am in and around the predominantly Sunni Arab town and continued for several hours, a tribal leader said, with nine other people wounded in the firefights and bombardment.
Among the dead were two children, according to Dr Ahmed Shami, the chief of Fallujah's main hospital. Another two children were wounded.
In a sign of both the reach of anti-government militants and the weakness of security forces, all of Fallujah and shifting parts of Anbar provincial capital Ramadi, farther west, have been out of government control since early January.
Militants subsequently seized parts of Ramadi and all of Fallujah, the first time anti-government forces have exercised such open control in major cities since the peak of the deadly violence that followed the US-led invasion of 2003.
The bloodshed comes during vote counting from the April 30 general election, the first since American troops withdrew in late 2011, and amid a protracted surge in nationwide unrest.