Moon said North Korea will face harsher global sanctions and pressure if it resorts to fresh provocations, adding that "denuclearisation of the Korean Peninsula is path to peace and our goal".
President Trump, who has repeatedly called Kim Jong-Un "Rocket Man" and his government a "band of criminals" while threatening to destroy the country with "fire and fury", said Thursday that he has a "very good relationship" with the Korean dictator.
"A lot of good talks are going on right now", Trump said Wednesday during a news conference at the White House with Norwegian Prime Minister Erna Solberg.
Putin's comments come after the two Koreas held talks on January 9 in an effort to ease tensions ahead of the Winter Olympics scheduled for February 9-25 in the South Korean city of Pyeongchang. "I want to show my gratitude", he said.
Speaking to Newsweek in December, Robert Einhorn, an expert in nuclear non-proliferation at Washington-based think tank Brookings Institution, said: "Trump's bombast was not just an embarrassment and a blow to the USA reputation for skilful diplomacy".
The US has not had any direct contact with North Korea since 2012, at least officially.
President Donald Trump said on Wednesday that the United States could "conceivably" return to the Paris climate accord, although he stopped short of signaling any move in that direction. I could give you 20 examples.
"Only the first part of the dialogue was open to press, but from what cameras caught, both sides were in upbeat moods and seemed earnest about making a deal at the outset", she tells Morning Edition.
The two Koreas released a joint statement saying, "South and North Korea have chose to make joint efforts for the unity of the people and reconciliation by establishing an environment for peace and easing military tensions on the Korean Peninsula".
"I believe it would greatly help ensure the success of the PyeongChang Winter Olympic Games if you could express an intention to delay joint South Korea-U.S. military exercises during the Olympics in case the North does not make any more provocations", Moon was quoted as telling Trump, Yonhap News Agency reported.
Trump's administration's official position on North Korea says that the "right circumstances" will occur when North Korea is willing to discuss denuclearization, something which statements from Pyongyang routinely refuse and bristle at.