Authorities scrambled a search and rescue operation overnight northwest of Paros. Air and sea assets were deployed to the area, the coastguard said.
A video released by the coast guard on Saturday (December 25) showed the migrants boat sunken beneath the sea.
The country's semi-official Athens News Agency said the bodies of 12 men, three women and an infant were recovered from the area. The 63 rescued were taken to shelter on Paros island, with some receiving first aid treatment.
Eighty people were believed to have been on the vessel, which according to the coastguard was believed to have been headed for Italy from Turkey.
The circumstances under which the vessel capsized were unclear.
Giannis Plakiotakis, Greece's shipping minister, said trafficking gangs were responsible for the disasters.
Earlier Friday, Greek authorities said they had recovered 11 bodies from the scene of another shipwreck, when a sailboat carrying migrants sank off the uninhabited islet of Prasonisi, near Antikythera island in southern Greece on Thursday. Another 90 people were rescued in that operation. The coastguard said initial information suggested those migrants had also been heading for Italy.
Overnight between Tuesday and Wednesday another boat thought to have been carrying up to 50 migrants sank off the island of Folegandros, with dozens feared missing.
Greece is one of the main routes into the European Union for migrants and refugees from Africa, the Middle East and beyond, though the flow has tapered off since 2015-2016, when more than a million people traversed the country to other EU states.