Thousands of people in Dhofar and the nearby al-Wusta province were evacuated as strong winds and torrential rainfall hit the coastal areas.
Authorities have told residents to stay inside and Salalah's airport has been closed until midnight on Saturday local time (20:00 GMT).
According to the authorities, Mekunu had been declared a Category 2 storm on Friday, with wind speeds that went up to 167kmph at its centre, but it has already started to weaken. "Its predicted path brings it very close to or just to the west of the city of Salalah, Oman's second-biggest, with a population of about 200,000", it added. More than 30 people were still missing in Socotra, including Yemeni, Indian and Sudanese nationals.
The 17 missing people had been in two boats and three vehicles that had been sunk or swept away by the storm, according to Socotra Governor Ramzi Mahrous.
The cyclone also hit areas in southeast Yemen neighbouring Oman.
Flash floods engulfed Socotra's streets, cutting electricity and communication lines.
The center of the storm is now about 85 miles south-southwest of Salalah, Oman - the third-largest city in the country, with a population of over 300,000 people. Some cars were left abandoned on the road.
The strongest on record to make landfall over the Arabian Peninsula was super cyclone "Gonu" in 2007 which hit North-East Oman with highest wind speeds recorded at 270 km/hr.
The port of Salalah, crucial to Qatar amid a boycott by four Arab nations over a diplomatic spat with Doha, said it also had taken precautions and secured cranes ahead of the cyclone.
A video shared by the Oman Observer news outlet showed flood waters overcoming the Wadi Sahalnoot damn near Salalah.
Police said a 12-year old girl died when a gust of wind sent her smashing into a wall.
Authorities relocated over 230 families to sturdier buildings and other areas on Socotra, including those more inland and in the island's mountains, Yemeni security officials said.
An IMD image from 24 May showing cyclonic storm Mekunu.
The island, listed by UNESCO as a World Heritage Site, has been the focus of a dispute between the UAE and Yemen's internationally recognized government amid that country's war after Shiite rebels, known as Houthis, seized the Yemeni capital, Sanaa.