Around the same time, fighters attacked another checkpoint manned by government-backed fighters from groups known as Awakening Councils in the mostly Shiite Shaab district in the capital's northeast.
One of the council members was killed and two were injured, police officials said.
Hours later, a bomb attached to a policeman's car killed the driver and wounded two passengers, who were also policemen, officials said.
Security forces have been frequent targets of attack in recent months as the United States reduces its troop strength in Iraq more than seven years after the US-led invasion that ousted then-leader Saddam Hussein.
The US plans to cut its troop numbers to about 50,000 by August, 31, the date for the formal end of combat operations, from just under 60,000 now.
But fighters appear to be trying to take advantage of a vacuum as Iraqi politicians squabble over positions in a new government more than five months after an inconclusive March 7 election.
Overall violence in Iraq has ebbed since the peak of sectarian warfare in 2006-07 but bombings and other attacks occur frequently.
Iraqi authorities said the number of civilians who died in bombings and other attacks nearly doubled in July to 396 from 204 in June.