Vanuatu's National Disaster Management Office (NDMO) has issued a red alert for the southern Tafea province, urging residents to stay indoors as Cyclone Ula passes the country's south.
The category-four storm is tracking near the southern islands after hitting Fiji and Tonga last week.
Peter Korissa from the NDMO said people in the southern islands of Aneityum and Futuna were experiencing strong winds.
At 8:00pm on Sunday (local time), the cyclone was 95 kilometres from Aneityum.
Ellen Luke from the Tropical Cyclone Warning Centre in Port Vila said it was moving south-west, away from Vanuatu.
"But people [in the southern islands] are still experiencing heavy rainfall and strong gale force winds," she said, adding that seas would also be rough as the cyclone passes.
She said the storm was expected to weaken to category-three strength later on Monday.
Care International's Vanuatu program director, Inga Mepham, said it was too early to know the extent of the damage.
"Local assessment teams will be deployed at first light to the islands of Anietyum, Futuna and Tanna. Only after reports start coming in will we begin to get a true picture of what the situation is," she said.
"Most people have spent the night sheltering in schools and churches, the only permanent buildings on these islands.
"We know there has been some damage to gardens and that people living in flood-prone areas have been evacuated."
Authorities in New Zealand expect heavy rain in the north from Monday, as the cyclone curves south-east.
The meteorological service in Vanuatu also predicted heavy rain and possible flooding in low-lying areas, places close to river banks and along the coast in Tafea province.
Last March Vanuatu was hit by Cyclone Pam, which left tens of thousands of people homeless and wiped out food crops.