South Korea's presidential office immediately convened a National Security Council meeting on the issue.
He also calls for a "united and clear reaction of the European Union".
"We strongly urge North Korea side to face up to the firm will of the global community on the denuclearization of the peninsula, abide by relevant resolutions of the UN Security Council, stop taking wrong actions that exacerbate the situation and are not in its own interest, and return to the track of resolving the issue through dialogue", the Chinese Foreign Ministry said in a statement.
Its fifth nuclear test in September 2016 was measured to be possibly North Korea's biggest detonation ever, but the natural disaster it caused was still not believed to be big enough to demonstrate a thermonuclear test. State-controlled media say it was a hydrogen bomb.
South Korea's meteorological agency called it a "man-made" natural disaster. It's calling for all parties to refrain from escalating tensions further. The North doesn't need to use its nuclear weapons - it has 20,000 conventional rocket launchers, artillery pieces and heavy mortars - to attack Seoul, which is less than an hour's drive from the inter-Korean border.
The North followed its two tests of Hwasong-14 ICBMs by threatening in August to launch a salvo of its Hwasong-12 intermediate range missiles toward the US Pacific island territory of Guam. The country's state-controlled media say it was a thermonuclear device and a "perfect success".
North Korea also claimed to have tested a hydrogen bomb in January 2016, but other countries, including the USA, doubted that claim.
The two leaders also agreed that pressure must be raised on North Korea, Abe told reporters.
China, North Korea's only real ally and global patron, has yet to respond.
China's foreign ministry said in a statement that the Chinese government has "expressed firm opposition and strong condemnation".
China urged North Korea to "stop taking erroneous actions that deteriorate the situation".
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The president's office would not comment on it means by "strongest strategic assets".
If confirmed as a nuclear test, this act would indicate that the DPRK's nuclear programme is advancing rapidly.
US President Donald Trump has warned Pyongyang that it faces "fire and fury", and that Washington's weapons are "locked and loaded".
Pyongyang, which says it needs nuclear weapons to defend itself, carried out its first atomic test in 2006.
North Korea's weapons program has been progressing at a rapid pace under leader Kim Jong Un. It said leader Kim Jong Un attended a meeting of the ruling party's presidium and signed the go-ahead order.
South Korean authorities said it appeared to be artificial, consistent with a nuclear test.
A significant earthquake's been registered near North Korea's nuclear site, raising fears it has carried out its sixth test.
NORSAR, an independent seismic monitor, estimated the blast created a yield of about 120 kilotons.
Japan's defense minister says the larger magnitude of the quake suggests "capability significantly exceeding the last one".
"If we look at it from Pakistan's example, the North might be in the final stages" of becoming a nuclear state, he said. That would be 3:30 p.m.in Seoul and Tokyo, and 2:30 a.m. EDT. The report could not be independently verified.
It is clear, however, that each new missile and nuclear test gives the North invaluable information that allows big jumps in capability.
The Korea Meteorological Administration said Sunday that it had not detected another quake.