South Korea's new president launched global efforts to defuse tension over North Korea's weapons development on Thursday, urging both dialog and sanctions while also aiming to ease China's anger about a USA anti-missile system.
Meanwhile, Chinese President Xi Jinping congratulated Mr Moon on his election, saying he would be willing to work with him on a basis of "mutual understanding and mutual respect".
The South is also embroiled in a dispute with China over a controversial U.S. missile defence system, which Beijing sees as a threat.
During the call, Moon told Xi that South Korean people and companies in China "are going through many difficulties", and said he hoped Xi could "pay special attention so the restrictions and sanctions could be smoothly resolved", Yoon said.
The THAAD deployment was agreed by Moon's predecessor, conservative leader Park Guen-hye.
Moon wants to increase dialogue and engagement with North Korea while also maintaining pressure and sanctions to encourage change.
Moon's spokesman said the presidents agree on the "common goal" of denuclearizing the Korean peninsula, and that Moon understands China's concerns about the U.S. Terminal High Altitude Area Defense (THAAD) system.
On Tuesday, Beijing confirmed that an official delegation from North Korea would also attend the summit.
Mr Moon's office said the new president also spoke with Vladimir Putin on Friday - with the Russian leader pledging to play a constructive role in resolving the North Korea crisis. "I will go to Beijing and Tokyo, and even to Pyongyang under the right conditions.".
"This is in the common interest of China and South Korea as well as overall peace and stability in the region", Xi said, according to the statement.
Moon's attitude during the call appeared to significantly relieve senior bureaucrats in Japan who were concerned he would take a tougher stance and be reluctant to cooperate with Japan and the United States in dealing with North Korea's provocations.
"The North Korean situation is a very, very unsafe one for South Korea, for Japan, frankly for China and for the rest of the world".
President Moon talked up Australian wine and beef and said he hopes bilateral trade will develop further.
The call lasted 30 minutes and was the first conversation Moon Jae-in had with a world leader since his inauguration on Wednesday, Yonhap reported.
The two countries' foreign ministries struck a deal to resolve the so-called comfort women issue on December 28, 2015, consisting of an apology by the Japanese prime minister and a 1 billion yen ($8.98 million) fund for the victims.