Moon began his duties earlier in the day by receiving a briefing about North Korea from Army Gen.
Moon told Xi during the 40-minute call that he plans to send a special delegation to Beijing for talks on both topics, Moon's office said in a statement.
South Korea's first liberal leader in a decade, Mr Moon also said he would "sincerely negotiate" with the us, Seoul's top ally, and China, South Korea's top trading partner, over the contentious deployment of an advanced United States missile-defence system in southern South Korea.
The Terminal High-Altitude Area Defense (THAAD) system, which would intercept attacks from North Korea, became operational in South Korea last week.
"Mun Jae-In. was elected to be the 19th-term "president" with 41 percent voting rate", it said, using the McCune-Reischauer transliteration system that is standard in North Korea.
China also hopes the new South Korean government attaches importance to China's concerns and takes steps to promote the stable and healthy development of ties, Xi said.
Moon told Xi that it would be easier to settle the missile-defense issue if North Korea wasn't engaging in provocations, the statement said. The date was not set for a potential visit by Moon. The system has angered Beijing, which says its powerful radars allow Washington to spy on its own military operations.
He also said trilateral cooperation between Seoul, Washington, and Tokyo is vital to solving the North Korean nuclear issue. They agreed the bilateral relationship would continue to go from strength to strength.
SEOUL-South Korea's Moon Jae-in opened his presidency by declaring his willingness to go to North Korea when conditions are right, in keeping with a campaign promise to seek dialogue with his country's nuclear-armed neighbor.
Moon also received similar congratulatory calls from Japanese and Indian leaders on Thursday.
"Analysts say he is unlikely to weaken sanctions or reduce diplomatic pressure, given the much higher risks presented by North Korea after a decade of nuclear tests and vast improvements in its missile technology", the Guardian writes, noting that even Moon's proposed meeting with Kim would likely require a pledge from Pyongyang to abandon its nuclear and missile programs. Moon told Abe that their countries must not let their hard histories hamper co-operation in dealing with North Korea's nuclear program.