The boy, said to be aged just 50 days old, was travelling with a group of migrants that became trapped in a mountainous area. They had been abandoned by human traffickers who are creating a “scene of despair” on Tilos, the island’s mayor says.
Seawater was used to make the baby’s milk as the group’s resources ran low, officials added.
The authorities were able to rescue the baby by sea, while others were taken away from the area by helicopter.
Local media reported the baby was safe and healthy. It is not clear how many people were part of the group.
Tilos mayor Maria Kamma revealed details of the rescue at the weekend with a post on Facebook – and told how human traffickers are creating a “scene of despair” on the island, claiming some refugees have been “badly beaten”.
She wrote: “A 50-day-old refugee child who was stranded on the mountain for three days, his parents made his milk with water from the sea.
“Once again the community of Tilos stands up to the circumstances.
“A pregnant woman trapped on the mountain, babies and children thrown to the rugged parts of the island, women injured, men badly beaten by the Turkish traffickers, make up the scene of despair experienced by the society of my island in the last months.”
The mayor added there had been “minimum” help from outside of the island as it deals with the people traffickers – but praised emergency teams and volunteers for their ongoing efforts.
Greece is one of the gateways into Europe for migrants from the Middle East, Asia and Africa.
But the country has been accused by aid groups of forcibly ejecting migrants at its sea and land borders, also known as “pushbacks”, which is an illegal practice.
In July, the Greek coastguard denied accusations by Turkey that it had forced migrants away from one of its islands.
Turkey said drone footage showed a Greek vessel pushing a boat carrying migrants away from Lesbos island and into Turkish territorial waters.