During the talks, the North Korean delegation said it would send an Olympic delegation, including officials, athletes, cheerleaders, journalists and others, South Korean Vice Unification Minister Chun Hae-sung told reporters, according to media footage from the border village of Panmunjom, the venue of the talks.
Trump's remarks have shelved his often-bellicose rhetoric on Kim Jong-un, North Korea's top leader.
Moon was quoted as saying by BBC: "I think President Trump deserves big credit for bringing about the inter-Korean talks".
Moon briefed Trump on his government's talks with a delegation from the North on Tuesday, which resulted in Pyongyang agreeing to send a delegation to the Winter Olympics in Pyeongchang next month, both sides said.
He said in a news conference later that the United States had problems with Pyongyang, but a lot of good talks were going on.
"We would like nuclear talks to occur", said Goldstein, adding that this is "a good first step in that process".
While Moon praised the U.S. and the global community for putting pressure on North Korea, the South Korean leader also expressed concern this "could raise tensions and bring about unintentional clashes".
The three-point statement, read aloud by Ri Son-gwon, North Korea's chief envoy to the talks, declared that the South and North would cooperate to make the Pyeongchang Winter Olympics a success.
Adm. Richardson declined to comment on whether military to military engagements between North and South Korea could ease global pressure on Pyongyang or whether the burgeoning talks could affect USA military operations in the Pacific. But U.N. Ambassador Nikki Haley later said Washington isn't changing its conditions on talks with the North. But his North Korean counterpart is said to have made strong complaints about the issue.
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.
Minister Cho Myoung-gyon said that he does not think that inter-Korean dialogue will weaken the global community's efforts to enforce sanctions and pressure on the North over its nuke and missile programs.
This week, there have been signs of a thaw in the relationship between North and South Koreans after officials met to discuss the North's participation in the Olympics. USA officials did, however, reiterate the military's public support for the political effort being led by the State Department to address the North's provocations and its nuclear ambitions.
Goldstein said the world should be "encouraged" that North Korea would participate in the Olympics.
Per the State Department, there are no plans for the U.S.to meet with North Korean officials over the course of the games.
He confirmed the presence of the North Korean delegation while noting that the details of their visit remain to be debated.
For now, Trump appears to approve of the bilateral talks between the two Koreas and suggested that the United States would also be open to the possibility.