Some of the migrants had to be plucked from the water in the dark after the eight-metre pleasure boat ran into heavy seas off the island's southeast coast, the officials said.
The youngest was just five months old and several were kept in hospital for treatment, including a young child suffering from hypothermia and dehydration.
Three of those rescued were arrested on suspicion of people trafficking after others on the boat claimed they had paid for their passage from Lebanon to Greece -- 2,000 euros a head for adults and 1,000 euros for children.
It said most of those rescued are originally from Tartus, a government-controlled province on the Mediterranean coast that is one of the safer areas of Syria.
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They had travel documents and were believed to have been seeking a better life in Europe.
The rescue took place not far from a British army garrison where 114 migrants who landed on the beach at a British airbase last month are being housed.
Both the army garrison at Dhekelia and the airbase at Akrotiri, from which Britain carries out strikes against the Islamic State group in Iraq, are British sovereign territory and the fate of the 114 remains unclear.
EU member Cyprus lies just 100 kilometres off the Syrian coast but has so far avoided a mass influx of refugees from that country's conflict like that passing through the Balkans to Austria and Germany.
In September, 115 refugees, including 54 women and children, were rescued from a small fishing boat that ran into trouble off the south coast.
The UN refugee agency says more than 2,500 people have died trying to cross the Mediterranean this year, many of them Syrian refugees.