The cyclone, which killed at least seven people on the Yemeni island of Socotra on Friday, has now weakened and been downgraded to a tropical storm.
Oman's state news agency said authorities had evacuated hospitals in Dhofar province and areas along the Yemeni border, and the police and army are on alert. "Two Indian cargo ships have gone missing, losing five of their crew members".
On Friday night, Cyclone Mekunu struck the second largest city of Salalah in Dhofar province with torrential rains and high tides. However, the identities of the rescued men were not immediately known.
Yemen's government has declared the island of Socotra a disaster zone. A Saudi-led coalition is supporting Yemen's government in the country's three-year civil war.
Winds of up to 170 km per hour have been reported and thousands of residents near coastal areas have been moved to shelters.
But Socotra has been spared involvement in the violence, which has claimed almost 10,000 lives since March 2015 and triggered what the United Nations has called the world's worst humanitarian crisis.
Tropical Cyclone Mekunu made landfall overnight Saturday in Oman as a Category 3 hurricane equivalent storm, dumping almost two years' worth of rain in a single day on one of the country's most populous cities and killing at least three people there after killing at least two others on the Yemeni island of Socotra.
Omani forecasters warned Salalah and the surrounding area would get at least 200 millimeters (7.87 inches) of rain, over twice the amount of rain this city typically gets in a year. Tourist beaches were littered with debris and foam from the Arabian Sea.
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As Mekunu barrelled overhead, the eye of the storm provided a moment's respite.
Earlier, dust storms, followed by heavy rainfall and lightning, hit various parts of the country. Three Asians who were wounded by the cyclone were rescued.
Civil defence said it had evacuated 10,000 people from schools and government buildings, mainly in the city of Salalah which has a population of 200,000. "Water entered the front yard of my house and was just a foot from threshold of my front door", Abdullah Al Mushaikhi, a resident of Salalah told The National. Strong winds knocked over street lights and ripped off roofs.
India's Meteorological Department said the storm packed maximum sustained winds of 105 - 111mph with gusts of up to 124mph. Cyclone Mekunu will be "extremely severe" when it crashes into the Arabian Peninsula. Several underpasses became standing lakes. Standing water covered roads and caused at least one auto to hydroplane and flip over.
Electrical workers have been trying to fix lines in the city while police and soldiers have been patrolling in 4x4s.
The rain caused widespread flooding across the usually arid sultanate, turning normally dry wadis turbid, flooding streets and homes and causing a number of fatalities.
Salalah's airport has been closed until midnight on Saturday, local time (20:00 GMT).
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.