States of emergency have been declared in the Coromandel Peninsula and Bay of Plenty, with landslips, flash flooding and downed power lines closing roads.
Some coastal areas have been evacuated, and local media say 15,000 homes are without power.
But Auckland, New Zealand's largest city, has been spared the worst.
Cyclone Cook killed one person when it swept through the Pacific islands of New Caledonia earlier this week.
In New Zealand, the storm made landfall in the Bay of Plenty at about 18:30 local time (06:30 GMT). It is then expected to move to the South Island early on Friday.
Forecasters have been warning of 5m (16ft) waves, storm surges and 150km/h (90mph) winds.