The two countries also confirmed that Moon and Trump shared four broad principles in dealing with the North Korean nuclear crisis, Yoon said.
Amid condemnation in Seoul, Tokyo, Washington and Moscow, a jubilant North Korean leader Kim Jong Un promised more nuclear and missile tests and warned that his country's weapons could strike the US mainland and Pacific holdings.
"The president said we are leaving open the possibility of dialogue with North Korea, but we should sternly deal with a provocation to prevent North Korea from miscalculating".
US Disarmament Ambassador Robert Wood said yesterday that China's leverage was key.
Uruguayan Ambassador Elbio Rosselli, the current council president, said after the meeting that members are united in condemning the latest missile launch as "a serious threat to global peace".
Its state-run KCNA news service alleged that it now has the ability to send a "large-size heavy nuclear warhead" across the Pacific following its test of a Hwasong-12 missile over the weekend. The Musudan is one of the most advanced missiles that Pyongyang has tested successfully.
A US Navy commander on Wednesday pushed for a "sense of urgency" over North Korea's nuclear and missile development, stressing that it even threatens Pyongyang's ally China and neighbour Russian Federation.
"We need to return to dialogue with the Democratic People's Republic of Korea, stop intimidating it and find ways to solve these problems peacefully", Putin was quoted as saying, using North Korea's official name.More news: North Korea ferry service docks in Russia on inaugural trip
The Security Council, which has imposed six rounds of sanctions on the North, discussed further possible actions at yesterday's meeting.
The United States, which has 28,500 troops in South Korea to guard against the North Korean threat, has called on China to do more to rein in its neighbour. Third, dialogue with North Korea is possible when the circumstances are right.
Communications were severed by the North in 2016, Lee said, in the wake of new sanctions following North Korea's last nuclear test and Pyongyang's decision to shut down a joint industrial zone operated inside the North. "We believe South Korea will bring clear measures to improve relations".
South Korean Unification Ministry spokesman Lee Duk-haeng told reporters the government's most basic stance is that communication lines between South and North Korea should reopen.
Envoy to USA will discuss THAAD deployment and North Korea; envoy to Japan held meeting on comfort women agreement Special envoys to the USA and Japan bearing a personal letter from President Moon Jae-in departed for Washington and Tokyo on May 17.
While some observers suspected that Trump and Moon would have conflicting attitudes towards the North Korean threat, there has been a certain degree of overlap with regard to their ideas and approaches.
South Korea's Yonhap News notes that Moon has promised to increase South Korea's defense budget from the current 2.4 percent of GDP to 3 percent. Under UN resolutions, North Korea is barred from developing nuclear and missile technology.