He also said his country will ready its military so it is prepared to carry out pre-emptive attacks, calling the current situation very precarious, according to the Korean Central News Agency.
The threats are part of the authoritarian nation's regular propaganda effort to show strength in the face of what it sees as an effort by its enemies South Korea and the United States to overthrow its leaders.
It follows harsh UN sanctions over the North's recent nuclear test and long-range rocket launch and comes ahead of joint US-South Korean war games this month that the North claims are invasion preparations.
Pyongyang is thought to have a handful of crude atomic bombs, but there is considerable outside debate about whether it is technologically able to shrink a warhead and mount it on a missile.
"The only way for defending the sovereignty of our nation and its right to existence under the present extreme situation is to bolster up nuclear force both in quality and quantity," KCNA said.
It stressed "the need to get the nuclear warheads deployed for national defence always on standby so as to be fired any moment".
On Thursday, North Korea fired six short-range projectiles into the sea off its east coast, South Korean officials said.
That happened just hours after the UN Security Council approved the toughest sanctions on the North in two decades for its recent nuclear test and long-range rocket launch.
The firings also came shortly after South Korea's National Assembly passed its first legislation on human rights in North Korea.
The North Korean projectiles, fired from the eastern coastal town of Wonsan, flew about 60 to 90 miles before landing in the sea, South Korea's joint chiefs of staff said.
North Korean state media had warned that the imposition of new sanctions would be a "grave provocation" that shows "extreme" US hostility against the country.
It said the sanctions would not result in the country's collapse or prevent it from launching more rockets.
The UN sanctions include mandatory inspections of cargo leaving and entering North Korea by land, sea or air.
There is also a ban on all sales or transfers of small arms and light weapons to the North, and the expulsion of North Korean diplomats who engage in "illicit activities".