Civil defence authorities on Thursday advised people in low-lying areas on the North Island's Coromandel peninsula to evacuate to higher ground.
The water forced 2,000 people to evacuate and flooded hundreds of homes.
The storm was expected to move south overnight on Thursday, reaching the capital, Wellington, on Friday morning.
Since the rainfall last week, grower body New Zealand Kiwifruit Growers (NZKGI) has been identifying severely affected orchards near the township of Edgecumbe.
Flights across the country have been delayed or cancelled, with Air New Zealand suspending its operations from Tauranga Airport on the North Island.
"I have never seen an event like this in the 12 years I have been a forecaster in New Zealand. this is not an event to be taken lightly", forecaster Lisa Murray said before the storm's arrival.
It added: "Soil is already saturated after last week's rain event, so, add the rain from the Tasman Low and then the wind and rain from Cyclone Cook and you have the ingredients for large scale damage such as fallen trees, landslips and rapidly-rising rivers and streams".
They have warned that it would bring a "phenomenal" amount of rain and wind, reported The New Zealand Herald newspaper, compared with Cyclone Debbie which was more spread out.
The storm is due to make landfall later in the day over Bay of Plenty.
The Ministry of Education said 137 schools and 133 early learning centres across the North Island had been closed, affecting more than 37,000 school students. Landslides, flooding and wind damage from 150kph gusts are expected and airlines warn of "significant disruptions" with flight delays and cancellations.
New Zealand weather officials have said that the cyclone will be the worst to hit the country since 1968.
Rain in Dunedin will be set in with the chance of some flooding - but at this stage we're optimistic it won't be at the extreme end, but be vigilant around streams and watch for surface flooding on roads.
The cyclone has already killed one person in New Caledonia after it swept through earlier this week.
A state of emergency has been declared in the Bay of Plenty and Thames-Coromandel regions, which are still suffering from the aftermath of ex-tropical cyclone Debbie.
The New Zealand Transport Agency urged motorists to stay off the roads if possible, saying conditions would be hazardous.