The earthquake, as well as several aftershock tremors, struck the Cook Strait Sunday evening, the Wellington City Council said in a release Monday.
The City Council said an inspection of 2,500 buildings in the central business district of Wellington found that only 35 buildings sustained minor external damage -- mostly broken windows and cracked or broken masonry.
However, 12 buildings were cordoned off due to concerns about falling masonry and glass.
"We do not want to close Featherston Street to pedestrians and traffic -- but people in a hurry through the central city should avoid the street until further notice," Wellington Mayor Celia Wade-Brown said.
Key said the damage was not "tremendously significant," The Dominion Post reported.
"The city [Wellington] has come through it very well, from what we can see," Key said.
"There won't be anybody that has suffered damage where a legitimate claim ... won't be paid out ... . The advice that we've just had ... is that it doesn't look like the damage is tremendously significant. This is not a major financial event, at this point."
Meanwhile, some Wellington residents rushed to supermarkets and gas stations to stock up on supplies following the quakes, the Post said.
"The panic buying reports are around demand for water and candles," BP spokesman Jonty Mills said. "Trucks are still delivering and terminals are still open and we're monitoring every couple of hours."