During his call with Japanese Prime Minster Shinzo Abe, the two leaders agreed that their countries must not let their hard history hamper co-operation in dealing with North Korea's nuclear program, Moon's office said.
Moon wants to increase dialogue and engagement with North Korea while also maintaining pressure and sanctions to encourage change.
Liberal candidate Moon Jae-in has expressed confidence of winning office as South Koreans vote for a new president.
President Moon replaced the former president Park Geun-Hye who was ousted after a scandal broke last November, leading to her impeachment and ultimately her imprisonment this year.
"A candidate from the Saenuri Party was elected president in the south with a razor-thin margin", that dispatch read in full.
Japan has been concerned that Moon will take a tough line on feuds stemming from its 1910-1945 colonization of the Korean Peninsula and could fray ties at a time when cooperation over North Korea is vital. That played into the hands of Moon, who has written that South Korea should "learn to say no" to Washington. Conservative critics have anxious Moon's rise to power might cause a friction with Trump, who wants to increase pressure on North Korea with the help of China, the North's largest trading partner and aid benefactor.
With only an acting president in place for several months, South Korea has found itself caught between the US and China on the diplomatic and economic front.
The Iranian president also wished his South Korean counterpart and nation success and prosperity.
There's widespread opposition in South Korea to the THAAD deployment and loud protests from China, which also sees the system as a security threat.
Moon later spoke with Japanese Prime Minister Shinzo Abe and agreed to hold a bilateral meeting soon, Seoul and Tokyo said.
Ties between Seoul and Beijing have soured over the South's deployment of a controversial USA anti-missile system aimed at guarding against threats from the nuclear-armed 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.
Moon's message here about the need for negotiations is sure to please Beijing, since it largely echoes the Chinese position.
The conversation is a sign of willingness to fix relations between the two countries.
"Will Mr. Moon pursue a "pro-North, anti-Japan" stance?" the conservative Yomiuri newspaper said in an editorial Wednesday.
North Korea says it needs its weapons to defend itself against the United States which it says has pushed the region to the brink of nuclear war. "We hope the new administration values cooperation with Japan and the USA and develops realistic security and foreign policies".