U.S. State Department spokeswoman Heather Nauert welcomed Tuesday's Korean talks, which she said were "aimed at ensuring a safe, secure and successful" Olympics.
Ri added that Pyongyang's nuclear programme was not an issue between the North and South.
While Moon praised Trump and the global community for putting pressure on North Korea, leading them to reach out for talks, he also warned that the pressure "could raise tensions and bring about unintentional clashes".
North Korean officials suggested sending an Olympic delegation of athletes, high-level officials, performers and reporters to the global sporting event and Seoul assured it would meet the North's needs.
Among the issues to be sorted out is the number of athletes and officials from North Korea who will attend the Olympics from February 9-25, which is complicated by the fact that the deadlines for registration have already passed.
He cited the North Korean officials there as saying its delegation would include officials, athletes, cheerleaders and journalists. We have said, from the diplomatic side, we are ready to talk any time North Korea would like to talk.
"He would agree to a meeting if there were prospects for tangible results, " Moon said. The opening and closing ceremonies will be held at the venue appropriately known as Pyeongchang Olympic Stadium. South Korea proposed resuming reunions of families separated by the Korean War, but that offer was not included in the joint press release.
South Korea's unification minister on Wednesday dismissed concerns that Seoul's push for dialogue with North Korea may create a fissure in the united front pressing Pyongyang into giving up its nuclear weapons.More news: Despite deep ties to Vikings, Cameron Jordan a Saint first and foremost
Tuesday's talks, the first between the two Koreas since 2015, were held in the three-storey Peace House on the South Korean side of Panmunjom truce village that lies between North and South.
Early in the peace process, Trump criticized Moon for appeasing Kim, who as North Korea's dictator has sustained a brutal regime.
"It is very deplorable to see the US politicians boasting of their diplomatic failure as 'diplomatic success, '" it said.
Moon said North Korea will face harsher worldwide sanctions and pressure if it resorts to fresh provocations, adding that "denuclearization of the Korean Peninsula is path to peace and our goal".
"I came here with hopes that the two Koreas hold talks with a honest and faithful attitude to give precious results to the Korean people who harbor high expectations for this meeting, as the first new year present", Ri Son-gwon, said at the start of the talks as quoted by Yonhap.
But the USA has said that it must stop nuclear tests for talks with Washington to take place.
"For this reason, the DPRK side has chose to send a high-level delegation and those of its National Olympics Committee, athletes, cheerers, artists, other visitors, Taekwon-do demonstrators team, and journalists".