A ministry official, who asked not to be named, said about 20 people had been killed or wounded, in the first explosion.
The blast in the Iraqi capital's Adhamiya district on Tuesday badly damaged the ministry building and destroyed half a dozen vehicles. Smoke poured from the building as firemen fought a blaze sparked by the blast.
An Iraqi journalist told Aljazeera that the driver of the explosives-laden vehicle could not reach his target due to high security procedures including checkpoints.
The car then exploded a distance away from the ministry, the journalist said.
The body of one elderly man lay in the street on fire. Two of the dead were women.
The bomb exploded in a side street, close to a wall of the ministry building, at about 9:30am (0630 GMT).
Water from burst mains also flooded the streets.
Parents of children at a nearby girls' school raced to the scene in panic searching for their daughters. There were no reported casualties at the school.
There have been several car bomb
attacks in Iraq in the past week
US troops and Iraqi National Guards arrived later as American helicopters clattered overhead.
Neighbours said they had urged the ministry's security officials in the past to block off the side street where the bomb went off. "They didn't listen to us," said one man.
There was no immediate claim of responsibility for the attack, the latest of many bombings since last year's US-led invasion.
A second bomb targeting an Iraqi National Guards patrol near the Abu Ghraib prison killed four people, Aljazeera has learned.
Twelve others, mostly citizens on their way to the prison to visit detained relatives, were injured.
A National Guard who was in the convoy but was not hurt said US helicopters had taken away a dozen casualties.
The blast came after a night of US bombardment of the Iraqi city of Falluja.
Officials at Falluja General Hospital said early on Tuesday that six injured had been brought in overnight.
The US is doubling its troop
strength in neighbouring Ramadi
The latest wave of shelling occurred as American troops continued preparations for a major offensive against the city, a mainly Sunni settlement about 65km west of Baghdad, which has been resisting US-led control.
The order to launch the assault must come from Prime Minister Iyad Allawi, who warned on Sunday that his patience with negotiations was wearing thin.
Allawi has given no deadline for an attack on Falluja but has insisted that the city must hand over foreign fighters and permit government forces to assume responsibility for law and order.