Moon's administration also needs to figure out how to ease tensions with China stemming from the deployment of a US anti-missile defense system in South Korea to guard against the threat of a North Korean missile strike.
"For a long time, China has upheld the goal of the denuclearization of the Korean Peninsula, protecting the peace and stability of the peninsula and resolving the problem via dialogue and consultation", Xi said.
"If needed I will fly directly to Washington. If the conditions are right, I will go to Pyongyang", Moon said.
Nonetheless, aside from wanting to restore normal relations and trade with North Korea, Moon wants to have the same with China.
Just a day after taking office, there was already a different tone from Moon towards Japan compared with his impeached predecessor Park Geun-hye.
The corruption scandal that ended Park's presidency has subsequently dealt a blow to all her policies and in the course of his campaign Moon had promised a departure from approaches representative of the previous administration.
"I look forward to meeting H.E. Mr. Moon Jae-in soon and also look forward to working closely as special strategic partners", Modi added.
"President Moon said he understands China's interest in the THAAD deployment and its concerns, and said he hopes the two countries can swiftly get on with communication to further improve each other's understanding", Yoon told a briefing.
Mr Moon, 64, said: "I will urgently try to solve the security crisis". Critics say the North used the Sunshine Policy to take advantage of Seoul's generosity.
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Some analysts believe North Korea refrained from major provocations recently so as to not hurt Moon's chances in Tuesday's South Korean election. This could most easily be done with their common ground over THAAD, as China has complained that they don't like the idea of United States missile defense being deployed just off their coast.
The South Korean statement said Trump said North Korea's nuclear problem was hard, but one that could be resolved.
Meanwhile, the new president, after being sworn in has expressed his willingness to visit Pyongyang in the middle of high tensions with North Korea, an AFP report said.
Japanese Prime Minister Shinzo Abe is eager to turn a new leaf and focus on the future of bilateral relations instead.
Moon's election could add volatility to relations with Washington, given his questioning of the deployment of the USA military's Terminal High Altitude Area Defenses (THAAD) anti-missile system, but is not expected to significantly change the alliance, a US official said.
Mr Moon and the US President agreed "on close cooperation" to deal with the North's nuclear programme in a telephone conversation on Wednesday.
A spokesman for the new president said his government would "review of the validity of the decision" which saw THAAD deployed.
Conservative U.S. media voiced fears of potential friction between the two allies in dealing with a belligerent North Korea following nearly a decade of conservative rule in South Korea that led to a hard-line approach toward the North.
Xi told Moon that Seoul and Beijing should respect each other's concerns, set aside their differences, seek common ground and handle disputes appropriately, China's foreign ministry said in a statement.