New Zealand, a non-permanent member of the Security Council, holds the rotating presidency in September.
North Korean leader Kim Jong-un faced worldwide condemnation on Friday for what was believed to be a 10-kilotonne blast at the Punggye-ri nuclear site.
The US, Russia and China, often divided in the Security Council, were united in their response condemning the test.
The two countries most in danger of a nuclear-armed North Korea, South Korea and Japan, called for fresh measures against the regime.
South Korea's UN ambassador Oh Joon described North Korea's actions as "insane" provocation and spoke of his nation's need to build up its defences.
"As North Korea is saying publicly, they now have nuclear bombs mounted on missiles that can destroy our cities and they are proving it by all kinds of tests. It doesn't take rocket science to understand we are not left with too many options," the ambassador said.
"We should be able to defend ourselves against flying nuclear bombs or we should be able to make North Korea give up its nuclear programme."
US President Barack Obama said he would do what was necessary to protect the US and allies from North Korea.
"As commander in chief, I have a responsibility to safeguard the American people and ensure that the United States is leading the international community in responding to this threat and North Korea's other provocations with commensurate resolve and condemnation," he said.
- AAP