"We're not looking to the total annihilation of a country, namely North Korea", he said.
South Korea's President Moon Jae-in and Japanese Prime Minister Shinzo Abe have agreed to pursue stronger United Nations sanctions against North Korea.
Trump convened a White House meeting on Sunday afternoon of military leaders, his national security team and Vice President Mike Pence, where Mattis said they reviewed each of the United States' military options in the Asia-Pacific.
Theresa May, British prime minister, said: "I discussed the serious and grave threat these unsafe and illegal actions present with President Abe in Japan this week and reiterate the call we jointly made for tougher action, including increasing the pace of implementation of existing sanctions and looking urgently in the UN Security Council at new measures".
South Korea's presidential office also said Eui-yong spoke with USA counterpart H.R. McMaster for 20 minutes in an emergency phone call about an hour after the detonation.
It would not be the first time North Korea exaggerated its achievements - they wrongly claimed to have tested an H-bomb last January - but the country is not only on the right path towards achieving its ambitions, but it is doing so far quicker than anyone expected.
"It is significantly hurting regional and worldwide peace and stability", he added.
North Korea on Sunday announced its sixth nuclear test, which seems to be its most powerful yet.
We will work with China.
"It also underlines yet again the urgent need for the global community to act on putting in place a legally binding ban on nuclear testing once and for all".
Meanwhile, the U.N. Security Council set up an emergency meeting for Monday at the request of the U.S., Japan, France, Britain and South Korea.
The Japan Meteorological Agency also observed a magnitude-6.1 tremor in North Korea, which showed a different waveform from a natural quake.
Russian President Vladimir Putin was in China for an economic summit and discussed the test with Chinese President Xi Jinping, the Kremlin said. But he summed up North Korea's rapid progress with a dire warning.
The leader chose not to address the test during his more than 40-minute address to the assembled leaders of Russia, India, South Africa and Brazil.
The urgent discussions were called after Pyongyang's latest nuclear test, with the rogue nation saying it successful trialled a hydrogen bomb that could be loaded onto a long-range missile.
"This is a shocking, dangerous, provocative, illegal regime that is threatening the peace and security of the region and the world and is advancing nobody's interests other than the maintenance of that one family's dictatorship of North Korea".