Greece has flown 30 refugees to Luxembourg in the first round of relocations under the EU's plan to redistribute migrants.
The six families, from Syria and Iraq, are among the first to be moved under aims to share out nearly 160,000 migrants across the bloc. Some EU members fiercely oppose the scheme.
More than 600,000 people have arrived in Greece by sea so far this year.
Meanwhile German police have launched raids on human trafficking networks.
More than 500 officers searched homes and businesses in the states of North Rhine-Westphalia, Lower Saxony and Baden-Wuerttemberg on Wednesday morning.
Germany expects at least 800,000 asylum seekers this year - some estimates are as high as 1.5 million. That is at least four times the number who arrived last year.
Greek Prime Minister Alexis Tsipras, who was at Athens airport to see off the first group of refugees on Wednesday, said they were making "a trip to hope".
But he warned they were merely a tiny fraction of the hundreds of thousands of people who have arrived on European shores this year.
A monthly record of 218,394 migrants and refugees reached Europe by sea in October, the UN says, almost as many as the total number of arrivals in 2014.
Most arrive in Italy and Greece after making a perilous journey across the Mediterranean.
The EU's migrant relocation scheme got under way last month, and at least 86 Eritreans and Syrians have already left Italy for Sweden and Finland.
A Greek asylum service official said the group leaving Athens on Wednesday, chosen because they were considered vulnerable, were made up of two Iraqi and four Syrian families, with a total of 16 children, according to AFP news agency.
On arrival in Luxembourg the families will reportedly spend two days in a registration centre and then two or three months in temporary accommodation as the authorities help them to find a permanent home, schools and jobs.
Most of those arriving in Europe head for Germany, but the government there has been calling on other EU countries to share the burden.
Under the EU's plan, refugees will be relocated to other European countries after being processed at so-called "hotspots" in Italy and Greece.
Until now migrants have made their own way on foot and by bus and train through the Balkans to Western Europe.
The Czech Republic, Hungary, Romania and Slovakia object to the relocation plan, urging the EU instead to reinforce the bloc's external borders.
Thousands of people have died while trying to make the journey to Europe, including around 450 people who have drowned on their way to Greece.
Five bodies have been recovered from the water after a boat carrying dozens of migrants ran into difficulty off the Greek island of Lesbos.