Briefing reporters ahead of the annual United Nations meeting, Haley and H.R. McMaster - the White House national security adviser - took a tough line on North Korea, warning that a military option to deal with its nuclear threats was available.
Trump called Kim Jong-un a rocket man as North Korea recently launched many ballistic missiles and also conducted its most powerful nuclear test.
The meeting will be held during the annual General Assembly gathering of world leaders at the United Nations where Trump will meet with the leaders of Japan and South Korea on the sidelines to address the crisis.
Standing before a B-2 stealth bomber and a vast United States flag, Donald Trump on Friday lashed out at North Korea warned that advanced USA weaponry could make the souls of America´s enemies "crumble".
Trump has said the USA and its allies will not be intimidated by the reclusive nation.
Kim had told his top officials that the goal of Pyongyang's efforts "is to establish the equilibrium of real force with the U.S. and make the USA rulers dare not talk about military option", according to the nation's state media.
Moon and Trump agreed it was important to thoroughly implement U.N. Security Council resolutions against Pyongyang, including the latest sanctions resolution unanimously adopted at the start of last week to punish the North for its latest nuclear provocation.
If diplomacy fails, she says Defense Secretary Jim Mattis "will take care of it".
"For those who have said and have been commenting about the lack of a military option, there is a military option", he insisted.
Many nations have put pressure on North Korea but that did not deter it from completing its actions.
"The leaders shared the importance of implementing the U.N. Security Council resolution under close cooperation with the global community and agreed to continue their discussions on the issue", the Cheong Wa Dae spokesman said.
"I believe it will further North Korea's isolation - diplomatic and economic isolation - because more and more nations are realising there's simply no collaboration with the worldwide community".
Cheong Wa Dae officials earlier said Moon and Trump will be joined by Japanese Prime Minister Shinzo Abe in NY for a three-way summit.
Trump, who took office in January, will emphasize the need for states to promote peace and prosperity, while upholding sovereignty and accountability as indispensable foundations of the global order, he said.
In Washington, the White House also provided a summary of the phone conversation, noting it was the second time North Korea had fired a missile over Japan in less than three weeks.