The storm struck as a category 3 system when it hit yesterday but has since weakened while moving away to the south.
The alert has been lifted and people are allowed out of their homes and shelters to assess the damage. Firefighters are now out clearing debris from the island's roads.
Power has been restored to some towns in the north.
Cyclone Cook crossed the main island between Houailou and Moindou.
The storm's centre passed about 100 kilometres to the west of Noumea, sparing the territory's most densely populated areas.
Meteo France said in Thio, on the east coast, a total of 350mm of rain fell in 24 hours.
The south of the main island was still being lashed by the storm and people are advised to be cautious.
Thousands of homes lost power when the cyclone made landfall yesterday.
Meteo New Caledonia forecaster Virgil Cavarero told Reuters the category three cyclone had been predicted to hit as a category four, which would have been even more devastating.
Heavy rain, gales and high seas around New Caledonia will gradually ease tomorrow as it moves away.
Meanwhile, Oxfam said flooding from Cyclone Cook has wiped out whole food gardens in low lying areas of Port Vila in Vanuatu.
Cook brought widespread flooding to the archipelago at the weekend.