The ballistic missile was reported to have been launched at the highest angle so as not to affect the security of neighbouring countries.
USA officials had yet to react to the North's claim, but Japan said, regardless, the new test showed "some progress" in the isolated country's missile programme and was thus further evidence that "we have entered a new level of threat regarding North Korea".
The statement stresses the importance of working to reduce tensions and maintain peace and stability on the Korean peninsula.
The statement also revealed that, "The members of the Security Council vowed to fully implement all measures imposed on (North Korea) by the Security Council and strongly urge all other member states to do so in an expeditious and serious manner".
Despite North Korea's claim that Sunday's test simulated a re-entry situation, South Korean defense officials say the North probably has yet to master the technology.
The Security Council has approved six increasingly tougher sanctions resolutions following its nuclear and missile tests.
China Foreign Affairs University global relations professor Su Hao said North Korea had "to be held accountable and pay the price" for violating United Nations resolutions. For Pyongyang, this would mean a guarantee that the Kim regime survives and that its nuclear program can continue.
Council diplomats say the USA and China, the North's closest ally, have been working on a new sanctions resolution.
Russian President Vladimir Putin weighed in Monday morning, calling the test "counterproductive" and "dangerous", but he also suggested the USA and it's regional allies could bear some of the blame, saying "we must stop intimidating North Korea".
A closed-door emergency Council session was set for Tuesday.
U.S. Ambassador to the United Nations Nikki Haley also condemned the test, saying it "is not the way to sit down" with President Donald Trump.
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He suggested they were "two different forms of entertainment" and that Netflix "brings a great amount of connectivity". It should not be part of your everyday setting", said the Spanish filmmaker.
"I think this is an important step sort of showing where it is headed", Wright said.
North Korea launched another ballistic missile on Sunday, which it said was aimed at verifying the capability to carry a "large-scale heavy nuclear warhead", the latest in a series of tests that has increasingly alarmed its neighbors and raised tensions with the United States.
China might support a new round of United Nations sanctions against North Korea after Pyongyang launched another missile test just hours before Beijing held its biggest diplomatic event of the year, the belt and road forum, observers said.
Han said North Korea's missile development was progressing faster than expected.
North Korean leader Kim Jong-un.
Even before North Korea gave its account of what happened, the launch caught the eye of experts.
Japan's Prime Minister Shinzo Abe told reporters that the launch was "absolutely unacceptable" and that Japan would respond resolutely.
But Moon slammed the latest launch as a "reckless provocation" and said dialogue would be possible "only if the North changes its attitude".
"They have the overwhelming dominant economic relationship with North Korea and because they have the greatest leverage, they have the greatest responsibility", he added.
North Korea claimed the missile is capable of being armed with a nuclear warhead.