The surprise test was personally ordered by North Korean leader Kim Jong-Un and came just two days before his birthday.
If confirmed, the test would mark another big step toward Pyongyang's goal of building a warhead that can be mounted on a missile capable of reaching the US mainland.
The US said it was too early to verify North Korea's claim but vowed to "respond appropriately" to "any and all provocations."
Japanese Prime Minister Shinzo Abe said, "The nuclear test that was carried out by North Korea is a serious threat to the safety of our nation and we absolutely cannot tolerate this."
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"This clearly violates UN Security Council resolutions and is a grave challenge against international efforts for non-proliferation," he said, adding his country would seek to coordinate efforts among UN members to deal with the action.
She said North Korea's action is a strong challenge to international peace and stability.
"Tough measures are needed, including the strict sanctions of the UN Security Council and allied and friendly nations," she was quoted as saying by Yonhap news agency.
China, North Korea's key ally, "firmly opposed" the test which was conducted "irrespective of the international community's opposition".
"We strongly urge the DPRK side to remain committed to its denuclearisation commitment, and stop taking any actions that would make the situation worse," foreign ministry spokeswoman Hua Chunying told a regular briefing, using the North's official name.
The EU condemned North Korea's claimed hydrogen bomb test as a "grave violation" of a UN ban on its development of nuclear weapons and a threat to the region.
EU foreign affairs head Federica Mogherini in a statement asked North Korea "to cease this illegal and dangerous behaviour" and to return to what are known as the Six Party talks - grouping of North and South Korea, China, Russia, the United States and Japan - on Pyongyang's nuclear programme and the future of the divided Korean peninsula.