The 115 people, including 54 women and children, had been aboard a small fishing boat about 40 nautical miles from the southern port of Larnaca at the time they struck trouble, said a source in the island's Joint Rescue Coordination Centre.
Cyprus police said they had arrested three people who had admitted to smuggling the refugees, while a fourth was on the run.
Hours after they were rescued, more than half of the refugees -- who included some Lebanese and Palestinians -- were taken by bus to a reception camp at Kokkinotrimithia outside Nicosia.
"We didn't want to come to Cyprus. Life is too expensive here and it is difficult to find a job. This island is for tourists," said Zeina Joseph, a 38-year-old Palestinian who lived in Syria.
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"We were at sea but didn't know where exactly we were," she said, adding that her journey had cost her $3,500 dollars and lasted three days.
Muwaffaq Abu Jeish, another refugee, said: "I saw death with my own eyes, and it was a slow death."
"The migrants want more. They want, like many others, to go to northern Europe, specially that today they see so many who have succeeded in reaching Germany," said reception camp director John Avlonitis.
Germany, Europe's biggest economy, has taken in by far the EU's largest numbers of migrants amid the biggest refugee crisis to hit the continent since World War II.
The total number is expected to reach 800,000 this year. This weekend alone, some 17,000 migrants were expected to have passed through Bavaria in southern Germany.
EU member Cyprus lies just 100 kilometres (60 miles) off the Syrian coast but has so far avoided a mass influx of refugees from that country's conflict, with most preferring to bypass the island.