The call to Moon took place just a day after he took office and comes amid increased regional tensions over North Korea as well as the deployment of a U.S. missile-defense shield in South Korea.
Speaking during his formal oath-taking ceremony on Wednesday, Moon pledged to work for peace on the Korean Peninsula amid growing worry over the North's expanding nuclear weapons and missiles programme.
"While it will be important to resolve the issues peacefully and diplomatically, holding a dialogue for the objective of a dialogue would be meaningless", Abe said, referring to Pyongyang's ballistic missile launches and nuclear tests.
Moon was sworn in as president of South Korea on Wednesday, a day after he was declared the victor of the election.
In response, Moon said he would like to exercise wisdom to resolve various issues so the two East Asian neighbors can develop a good relationship.
China sees it as a threat to its security and has called for its withdrawal, but has also denied it was doing anything to retaliate against South Korean businesses.
Putin said he was ready to help in all of the matters they discussed and the two leaders invited each other for state visits, the Blue House added.
Moon and Trump may also disagree on the US missile defense system THAAD.
"How to respond to North Korea.is an urgent issue".
Yoon Young-chan told a press conference here on Thursday that President Moon would dispatch separate delegations to discuss THAAD and the DPRK's nuclear program with their Chinese counterparts, although the two issues will be dealt with together. He has taken a more conciliatory line with North Korea than his conservative predecessors and advocates engagement.
On his second day in office, President Moon Jae-in spoke on the phone with the leaders of South Korea's two most important neighbors: China, followed by Japan.
Moon said in his first speech as president he would begin efforts to defuse security tensions on the Korean peninsula and negotiate with Washington and Beijing to ease a row over a USA missile defence system being deployed in the South.
The history text books were set to be issued next year despite widespread criticism that the plan was Park's bid to affect future generations to have positive interpretations about her own father's military regime.
Being a liberal candidate from the Democratic Party, which has always preferred the "Sunshine Policy" seeking to promote dialogue with North Korea and improve relations through cultural and economic exchange, Moon had to portray a progressive image in dealing with North Korea during his presidential campaign.