96 people were rescued from the vessel, which was found west of the small island of Halki, near Rhodes.
The total number of those people on board was not immediately known.
Among the survivors, Greek authorities say they have identified three suspected smugglers, who have been arrested.
Many of those who were rescued, including children, had not been wearing life jackets.
The coast guard described the effort as “one of the largest and most successful search and rescue operations to have taken place in the Aegean Sea”.
The authorities were initially alerted to the sinking vessel when a passenger on board made an emergency call late on Tuesday.
"Four coastguard patrol boats, a rescue boat, a naval vessel, five cargo ships sailing in the area and an air force helicopter took part in the operation", according to a statement from the Ministry of the Merchant Navy.
Of those rescued, 72 were transported to Rhodes, five to the island of Karpathos, and 19 were picked up from a cargo ship by the Turkish coast guard.
The migrants to Greek islands were being given coronavirus tests and were being housed in quarantine hotels.
But Greece’s coast guard said its rescue efforts were hampered by the Turkish coast guard after two patrol vessels arrived overnight during the search when a “large number” of people had already been rescued.
There was no immediate response from the Turkish coast guard to the accusations.
The rescue comes during a crisis in Greek-Turkish relations over maritime exploration rights in the eastern Mediterranean.
Greece's coastguard says there have been at least 25 incidents in the last three months of migrant smuggling gangs bypassing the country to reach Italy.
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