They said emergency services were on hand to offer first aid, food, water and blankets to the 83 migrants after their boat was towed ashore in the northwest coastal area near Paphos.
Among those rescued from the drifting boat by the Cypriot coast guard were 37 children, 14 women and 32 men, said police. All were said to be in good health.
A pregnant woman was taken to hospital as a precautionary measure.
Police were questioning an adult member of each family and a reception center outside Nicosia was being prepared for the migrants.
The boat departed from Mersin in Turkey Wednesday evening and arrived in open waters off Cyprus early Friday when the captain abandoned ship due to engine trouble, police said.
The captain reportedly escaped in another boat, said the official Cyprus News Agency, which added that migrants said they paid between $3,000 and $10,000 each to people smugglers.
EU member Cyprus lies just 100 kilometers (60 miles) off the coast of war-torn Syria but has so far avoided a mass influx of refugees.
The Paphos coastal area, however, has been a target of traffickers operating from Turkey, In September, police located 50 Syrian migrants in the area.