George Voulgarakis, 46, and his family were at home at the time of the attack early on Tuesday, which also damaged four cars. Debris flew into nearby streets and an empty school playground.
"[The bomb] was placed under a car about 70 metres from his house. It went off just before he left for work," a police official said. "It was really strong, it was aimed at Voulgarakis."
The blast occurred at 7:42am on a residential street in the Kolonaki district in central Athens, the source said.
Neither police nor media received any warning calls before the explosion or any claims of responsibility, as often happens with such blasts, the police said.
They said the bomb had been detonated by a timer.
The area surrounding the home of Voulgarakis has been patrolled by police since a small explosive device went off in the spring of 2004.
Voulgarakis had been public order minister until February and his move to the culture ministry during a reshuffle was widely seen as a demotion due to a phone-tapping scandal that made the news in Greece for weeks.
Voulgarakis is despised by segments of Greece's political left, who accuse him of covering up the alleged kidnapping and interrogation of 28 Pakistanis in Athens by secret service operatives after the bombings in London last July.
The case of the 28 Pakistanis
The minister has consistently denied any such cover-up.
Large bomb attacks have become a rarity in Greece since the leaders of the November 17 urban guerrilla group were captured and convicted before the 2004 Athens Olympics.
The group killed 23 people in 27 years of drive-by shootings and bombings.
Several leftist and anarchist groups are still active, but their attacks usually cause no injuries.
Police said Voulgarakis was on a list of government officials considered possible targets.
"Orders were given to immediately step up controls at all the other possible targets," the police official said.