The Revolutionary Struggle group said it aimed to assassinate George Voulgarakis on May 30, but he was unharmed.
The group chose Voulgarakis because of his role in two government scandals when he was public order minister.
The remotely detonated bomb planted near the minister's home was intended to kill him as he was going to work.
It wrecked cars and shattered windows in central Athens but caused no injuries. Voulgarakis was still in his house.
"We wanted to kill Voulgarakis," the group said in a pamphlet sent to a Greek weekly newspaper, To Pontiki, which passed it on to police. The paper comes out on Thursday.
Voulgarakis, 46, had been public order minister until February and his move to the culture ministry during a government reshuffle was widely seen as a demotion due to his handling of a phone-tapping scandal.
A government spokesman, Evangelos Antonaros, said: "All democratic Greek citizens condemn unequivocally actions of raw violence against our society and democracy."
Revolutionary Struggle claimed responsibility for a bomb at the Economy Ministry in central Athens in December, which wounded two people and damaged buildings.
Voulgarakis was also the target of an attack in 2003, when he was part of the conservative New Democracy party in opposition and a gas canister bomb destroyed cars parked under his building.