The girl died after strong winds swept her against a wall, police say. Another person was washed away in a car.
State TV showed large areas of flooding in Dhofar and al-Wusta provinces.
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.
The second death in Oman was recorded after a car drifted into a valley in the southern region of Dhofar following torrential rain, Royal Oman Police tweeted on Saturday.
Thousands of people living near coastal areas in Dhofar and al-Wusta had been moved to shelters as 170km/h (105mph) winds and torrential rain hammered the region, officials said.
Residents were told not to take risks by crossing valleys in the region.
Cyclone Mekunu’s landfall to the west of Salalah, Oman’s second city, came a day after it lashed the Yemeni island of Socotra, where at least seven people were reported to have been killed and several more said to be missing.
While the storm has been downgraded, officials said it was still pummelling the country with heavy rain and high winds. Oman’s Directorate General of Meteorology said the storm was now heading north-west into Saudi Arabia.
In parts of southern Oman where the storm has subsided, clean-up operations are under way via security forces and emergency services.
Salalah’s airport has been closed until midnight on Saturday, local time (20:00 GMT).
Meanwhile, Yemen’s government has declared the island of Socotra a disaster zone.
Fisheries minister Fahad Kafin told AFP news agency the bodies of five Yemenis and two Indians had been recovered. Twelve Indians were still missing, he said.
Saudi troops were deployed to Socotra last month. A Saudi-led coalition is supporting Yemen’s government in the country’s three-year civil war.