"They agreed it would be important for the global community to continue to pursue a robust response to North Korean violations of UN Security Council resolutions and other worldwide obligations".
A US State Department spokeswoman said on Thursday that Washington remained open to talks with North Korea, but needed to see that it would "cease all its illegal activities and aggressive behaviour in the region".
Mr Moon has adopted a more conciliatory tone towards the North - backing diplomatic engagement and suggesting he could visit Pyongyang for talks.
Since the beginning of previous year the North - which says it needs atomic weapons to defend itself against invasion - has mounted two nuclear tests and a series of missile launches.
Chinese leaders don't often make congratulatory telephone calls to new foreign leaders right after they take office, normally sending envoys days or weeks later carrying personal letters instead.
"I hope Russian Federation will play a constructive role to help place North Korea back on the track to denuclearization".
But Abe also said "dialog for dialog's sake would be meaningless" and he called on North Korea to demonstrate "sincere and concrete action", Hagiuda said, adding that Moon shared Abe's views.
Moon officially took office on Wednesday after winning Tuesday's snap presidential election with over 41.08 percent of votes, according to South Korea's National Electoral Commission.
While South Korea, China and Japan all share worry about North Korea, ties between South Korea and China have been strained by South Korea's decision to install a US anti-missile system in defense against the North.
China has protested against the deployment saying the system's powerful radar can probe deep into its territory, undermining is security, destabilizing the regional security balance and doing little to deter North Korea.
The last time that Han appeared to brief foreign diplomats in Pyongyang was last December, to present North Korea's response to the latest round of United Nations sanctions after the September 2016 nuclear test.
Ties between South Korea and China, who are important trade partners, have been strained by Seoul's decision to host a U.S. Terminal High Altitude Area Defense (THAAD) anti-missile system in response to a growing missile threat from North Korea. Moon responded that he's aware of China's concerns and he hopes the two countries can have a greater understanding of each other's positions on the system, it said.
As well as clouding efforts to rein in North Korea's nuclear ambitions, the THAAD deployment has also led to recriminations from Beijing against South Korean companies.
"I, too, aim to find a way to begin talks quickly between North and South Korea as well as the six-party talks", Moon said, referring to talks aimed at denuclearising North Korea involving the United States, China, Japan, Russia and the two Koreas.
China has denied it is retaliating against South Korean businesses.