The leaders also called on the North to refrain from any more provocations, warning they will only lead to its isolation and eventual collapse, according to Park Soo-hyun, spokesman for South Korea's presidential office Cheong Wa Dae.
"The two leaders agreed to further strengthen the close cooperation between South Korea and the United States, and put more practical pressure on North Korea based on their cooperation to make North Korea realize it will only become more isolated diplomatically and face additional economic pressure if it continues to make provocations and eventually enter the path of collapse," Park said of the telephone conversation between Moon and Trump held earlier in the day.
The telephone conversation, the second of its kind since the start of the month, followed the North's launch of what appeared to be an intermediate range ballistic missile that flew over Japan on Friday.
Their first phone conversation this month was made shortly after the communist state staged its sixth nuclear test on Sept. 3.
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.
"The leaders shared the importance of implementing the U.N. Security Council resolution under close cooperation with the international community and agreed to continue their discussions on the issue," the Cheong Wa Dae spokesman said.
The South Korean leader underscored the importance of enhancing South Korea's own defense capabilities, along with those of the joint forces of South Korea and the United States.
Trump reiterated his full support for the Korea-U.S. alliance and said he would continue to provide any assistance and support necessary to further strengthen the alliance, Park said.
They agreed to continue their discussions when they meet in New York for the U.N. General Assembly later this week, according to the spokesman.
Cheong Wa Dae officials earlier said Moon and Trump will be joined by Japanese Prime Minister Shinzo Abe in New York for a three-way summit.
They said the South Korean and U.S. leaders were also expected to hold a bilateral summit, source from the Yonhap.