It said the child was on board a dinghy with 55 others, who said they were Syrians, when the vessel began to take on water as it approached the shore of Lesbos.
He was found unconscious by a coast guard patrol boat, and medics were unable to revive him.
The others on board were rescued unharmed. The ministry had no information on the boy's parents.
The coast guard also intervened twice during the night in the same region to rescue groups of 48 and 70 migrants on board boats that were taking on water, the ministry said.
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Some 800 of them crossed during the night, the most dangerous time of day.
More than 550,000 migrants and refugees have crossed into Europe by sea so far this year, most fleeing the civil war in Syria and the vast majority arriving in Greece, according to the UN refugee agency.
Some 3,000 people of them are dead or missing. Most of the victims in recent weeks have been children.