The daily sweep is needed to restore order to a country suffering from an "unprecedented invasion" that threatened the debt-racked nation's stability, Minister of Public Order and Citizen Protection Nikos Dendias said.
Police found 1,400 of the 6,000 immigrants detained during the weekend did not have proper documentation.
Dendias called the "invasion" the biggest Greece faced since the ancient Dorian invasion.
Some historians argue that around 1100 B.C. the Dorians, a Hellenic group, swept down from the north of present-day Greece, putting an end to the cultural period that is the historical setting of much ancient Greek literature and myth, including the epics of Homer.
The Dorian invasion has never been proved.
"Our social fabric is at risk of unraveling," Dendias told non-governmental Skai radio Monday. "The immigration problem is perhaps even greater than the financial one."
He threatened to resign if he was thwarted in the effort.
"There would be no point in me staying on," he said.
His ministry runs all security services, including the national police, coast guard, agrarian police and the National Intelligence Service.
The purpose of the operation, controversially codenamed "Xenios Zeus," is to restore human dignity to the immigrants, Dendias earlier told reporters.
"There are no racial criteria. We do not care about the color, nationality or religion of the illegal immigrants. Our only criterion is the observance of legality with absolute respect to the human rights," he said.
"Xenios Zeus" is one of the aspects of the ancient Greek god Zeus, sometimes called "Philoxenon," or "Friend of the Foreigner," and refers to Zeus as the patron of hospitality and guests.
Dendias said Monday the name was fitting because immigrants were living in unhealthy conditions, crammed into decrepit basement apartments with "no human rights" after being "conned by smuggling rings into believing they would be able to get a job and travel to Europe."
"Now they will return to their home countries. ... It's the best thing that could happen to them," the Green newspaper I Kathimerini quoted him as saying.
The minister said about 8,000 immigrants had already applied to leave the country under Greece's voluntary repatriation program.
Prime Minister Antonis Samaras of the conservative New Democracy party promised to crack down on illegal immigration in campaigning before the general elections in June.
Immigration from Africa, Asia and the Middle East has increased in Greece with the global economic crisis and Arab Spring revolutions, officials say.
Greece is on the European Union's southeastern border.
The Coalition of the Radical Left, or Syriza, which came in second in the June elections, denounced the "show me your papers" operation as an assault on human rights that fostered fear and racism.
The extreme right Golden Dawn party, which drew 7 percent of the vote, accused the government of not actually sending anyone back to a home country, merely holding a "badly organized PR stunt," Athens News reported.
Eighty-eight Pakistanis were flown back home on a chartered flight Sunday, a police spokesman told The New York Times. More deportations were expected soon, he said.
Greece has about 800,000 legally registered immigrants, while the number of those without papers is estimated at more than 350,000, I Kathimerini reported.