Haley said Trump had called Chinese President Xi Jinping and urged him to "cut off the oil from North Korea".
"That would be a pivotal step in the world's effort to stop this international pariah," she said, issuing a stern warning to Kim.
"If war comes, make no mistake: The North Korean regime will be utterly destroyed," she said.
The UN Security Council was meeting at the request of the US, Japan and South Korea to consider next steps after three rounds of sanctions adopted in the past year failed to push North Korea to change course.
France's UN ambassador Francois Delattre said the council should respond with a "tightening of the sanctions" — a move that would likely entail the adoption of a new sanctions resolution.
Japanese Ambassador Koro Bessho said "the international community has to keep the pressure up".
Earlier, Trump — who had traded barbs with Kim for months — had asked Xi to use "all available levers" to press the hermit state.
"Additional major sanctions will be imposed on North Korea today. This situation will be handled!" Trump said on Twitter.
There are concerns in Seoul that Trump might be considering military action against the North that could trigger a full-scale war.
Seoul is home to 10 million people and only about 50km from the border — well within range of Pyongyang's artillery.
North Korean state media said the missile launched on Wednesday was more sophisticated than any previously tested by Pyongyang.
"The ICBM Hwasong-15 type weaponry system is an intercontinental ballistic rocket tipped with super-large heavy warhead which is capable of striking the whole mainland of the US," the North's official news agency KCNA said.