Neither politician was at their office at the time of the explosion.
The car bomber turned off the main road and accelerated towards the checkpoint where the morning shift guards were gathered, Iraqi police said.
Fighters still manage to launch attacks in the Iraqi capital, although US and Iraqi officials say violence in Iraq, particularly in Baghdad, has fallen.
In the Saidiyah neighbourhood of Baghdad on Tuesday, a bomb hidden in a corpse blew up and killed an Iraqi soldier and wounded seven people, security officials said.
"The soldiers went to collect the corpse which was lying in an open area when the bomb inside it exploded and killed one soldier," an official said.
Meanwhile, the US military said American and Iraqi troops detained five men suspected of involvement in the assassination of the police chief in central Iraq's Babil province on Sunday.
Major General Qais al-Mamoori was near Babil's capital Hilla when a roadside bomb struck his convoy.
He was the third top Shia official to be assassinated in four months in provinces south of Baghdad amid intense rivalries between Shia groups.
Also on Tuesday, the US command in Iraq announced that a US soldier had died of wounds sustained when a suicide bomber blew up a car near the soldier's convoy in Salaheddin province north of Baghdad.
Two other soldiers were wounded in the blast, which occurred on Monday.