The agency was also urging caution on the Desert Rd, State Highway 1 between Waiouru and Turangi in the Central North Island, where up to 5cm of snow was expected on the road.
Heavy snow falls in Dunedin caused a number of accidents and closed roads in and out of the city.
Schools were also closed because buses could not take students to school.
Sergeant Ed Baker said Dunedin's roads had quietened down this afternoon once people realised they shouldn't be driving around in such conditions.
The Kilmog Hill, on SH1 north of Dunedin, was close to reopening but flurries of snow threatened to keep it closed, he said.
It had been so busy police were not attending minor accidents where cars slid off the road. There had been no serious injury accidents, he said.
Snow also closed Dunedin Airport this morning, cancelling a total of 10 flights and the travel plans of 480 people.
However, the snow cleared and flights were able to leave from 10.40am.
Dunedin Airport was only rarely closed because of snow, said Air New Zealand airport manager Alistair Bevin.
There was ice and snow on SH1 south of the city and it was blocked by a fuel tanker which jack-knifed, about 6am today at Lookout Point. The driver was not injured.
The MetService predicted Dunedin would experience slightly warmer temperatures tomorrow.
Snow showers would clear by evening for frosty night but it would be fine tomorrow for north and Central Otago.
Icy weather also caused problems inland.
The roads between Clinton and Balclutha and Milton and Balclutha are also closed due to poor conditions.
Gore probably hadn't seen snow like it for a couple of years at least, said Sergeant Stan Leishman.
There was about 5cm of snow on the flat and it was heavier on the hills, he said.
"It's zero degrees. The roads are freezing. There's black ice everywhere and the traffic in the main street of Gore appears to have very little traction at all."
Motorists needed to approach intersections at 10km/h to be able to stop in time and open road traffic was forced to travel about 60km/h, he said.
Mr Leishman said no road accidents had been reported yet but he was expecting plenty of them.
Invercargill experienced a frost on top of hail and sleet overnight, said Senior Sergeant Olaf Jensen.
Roads were all affected and driving conditions were treacherous, he said.
However, no serious crashes had been reported. There was a handful of minor crashes in rural areas where vehicles had slid off the road, he said.
There were a number of road closures and driving warnings around Southland and Otago.
The NZ Transport Agency is also reporting some roads require vehicles to use chains.
The roads include State Highway 6 - from Kingston To Dipton, State Highway 97 - from Five Rivers To Mossburn, State Highway 94 - from Gore to Mossburn and State Highway 8 - from Alexandra To Roxburgh.
Federated Farmers Southland president Rod Pemberton said snow was heavier inland but the weather was expected to pass through relatively quickly.
Rain or showers is forecast for most of the country today.
- NZ HERALD STAFF, NZPA
