"The logistical core of ETA has been detained today," with the arrest of those suspected of providing safe houses, stolen vehicles and false documents for members, Jorge Fernandez Diaz told a news conference.
He called it "a very important step towards the dismantling of ETA", which is classed as a terrorist group by the European Union and the United States.
The group is blamed for 829 deaths in a four-decade campaign of bombings and shootings for independence for the Basque Country of northern Spain and southwestern France.
It declared in October 2011 a "definitive end" to its armed activity but has not formally disarmed or disbanded as the Spanish government demands.
French authorities said the six were arrested in Blois, a city in central France, and Brive-la-Gaillarde and Montpellier, in the southwest of the
"Investigations determined that those detained operated in three groups that formed the core of ETA's logistical system," the minister said.
"The operation dismantled three services basic and vital to the survival of ETA and its members: the running of safe houses, the providing of stolen vehicles and the production of counterfeit material."
He said the two most important members detained on Tuesday were those who ran the safe houses: Antonio Goicoechea Gabirondo, 42 — an explosives expert — and Raul Aduna Vallinas, 32.
They were detained in Brive-la-Gaillarde. Ekhine Eizaguirre Zubiarre and Kepa Arkauz Zubillaga, both aged 29 and arrested in Blois, took care of forging documents, he said.
The other two, Igor Uriarte Lopez de Acuña, 39, and Julen Mendizabal Elezcano, 33, were in charge of stealing vehicles.
ETA has not carried out an attack since 2009 and experts estimate it has only a few dozen members still at large.
It has been weakened over recent years by a string of arrests of its members, many of them in France.
The Spanish government said 36 suspected ETA members have been arrested since the current conservative administration came to power in December 2011.
During the previous Socialist government's last term 342 were detained.
One of the group's top commanders got a life sentence in April for the murder of two Spanish police officers in France in 2007.
Spain refuses to hold talks with ETA's leaders.
"The operation, added to the extreme operational weakness of the terrorist group, places ETA face to face with the inevitability of its break-up," Fernandez said.
"The ETA we have known and suffered in Spain for decades will not exist again."
A source close to the case in Montpellier told AFP that two of the suspects will be transferred to Paris before the end of the week.
"Despite positive developments on this issue in the Spanish Basque country, our national territory is still a base for the logistical and military withdrawal of clandestine structures of the organisation," the French interior ministry said in a statement.