Powerful winds and torrential rain battered much of the island late Wednesday, toppling trees and damaging buildings.
More than 20,000 people have been affected by the storm, according to disaster management centre spokesman Pradeep Kodippily, who gave the death toll.
Two of the missing were on a fishing boat that capsized off the south coast.
President Maithripala Sirisena, who is currently on a visit to South Korea, ordered troops to help in the stricken districts. Military spokesman Brigadier Roshan Seneviratne said 2,000 troops had already been deployed.
Schools were closed Thursday across southern and central provinces and several flights into the main Colombo airport were diverted to Mattala airport in the south.
Power supplies were cut in many districts for several hours after the storm.