Vanuatu's weather bureau warned of damaging winds, rough seas and possible flash flooding as Cook passed over the archipelago's eastern islands and headed towards the capital, Port Vila.
The cyclone formed earlier than expected but continues on the forecast south-westerly track.
A long-range forecast shows the cyclone will hit New Caledonia's outer islands on Monday afternoon, and pass just north of the capital, Noumea, on Tuesday.
Local media reported gale-force winds over remote islands in the east of the Pacific archipelago, which were devastated by Cyclone Pam two years ago.
Neville Koop from the Nadraki weather service said the cyclone could be even stronger than predicted, judging by the recent experience of Cyclone Debbie in Australia.
"When that was in this same stage, we were thinking category one or two for Debbie," he said.
"But of course once it developed and we could see more about its structure and so forth, it became quickly very apparent that it could be stronger than that."
Vanuatu Meteorological Services said winds close to the centre were estimated at 75 kilometres per hour, and could increase to 110kph by the end of the day.
The disturbance comes at the end of the official cyclone season, which runs from November to April.
Parts of Vanuatu are still rebuilding after the category five Cyclone Pam struck on March 13, 2015.
It left 11 people dead and hundreds of thousands of others in need of food, water and shelter.