South Korean leaders see Pyongyang's participation in the Games as key to lowering tensions on the peninsula and possibly leading to diplomatic negotiations over the North's nuclear programme.
The Olympics have given a boost to engagement between the two Koreas after more than a year of sharply rising tension over the North's missile tests and its sixth and largest nuclear test in defiance of United Nations sanctions. The maximum pressure campaign must continue until North Korea denuclearizes.
On Sunday, North Korea called the news sanctions an "act of war".
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Pyongyang has frequently said it is willing to talk without preconditions, but Washington says it must first take concrete steps towards denuclearisation.
"The North Korean delegation too agreed that North Korea-U.S. relations must develop along with the South-North Korea relationship while noting (the North) has enough intention to hold North Korea-U.S. dialogue", he added.
Sunday's closing ceremony was attended by Moon, the North Korean delegation, and U.S. President Donald Trump's daughter, Ivanka Trump, among other dignitaries.
A Blue House statement said the North Koreans expressed willingness to engage in a dialogue with the US and that Moon stressed that talks must open as soon as possible.
The inclusion of Kim, a former spy chief who's on the list of individuals sanctioned by the US and South Korea, is considered controversial given his past ties to a fatal torpedo attack on a South Korean warship in 2010.
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The meeting was called following a Supreme Court ruling ordering the removal of Nawaz Sharif as party chief. Then the party's general council is required to elect a president within 45 days.
But the country has not shown readiness to enter talks on denuclearization.
His visit has enraged families of the sailors who were killed in the torpedo attack.
Choi's inclusion has raised suggestions there may be a meeting, however informal, between the USA and North Korea.
The eight-member delegation, led by General Kim Yong Chol, crossed the Demilitarised Zone yesterday morning.
The delegation told Moon during a meeting in Pyeongchang that North Korea was open to talks with Washington.
She was the center of a frenzy of attention, especially when she appeared at the opening ceremony and stood only a few feet from U.S. Vice President Mike Pence. They did not speak.
North Korea sent around 500 people to the Olympics, including officials, athletes, artists, journalists and 229 cheerleaders who amused spectators with their synchronized routines and songs. That, in part, is believed to have led to the North Koreans' cancellation of their meeting with the vice president.
Earlier a statement released by North Korean state media accused the United States of provoking confrontation on the Korean peninsula with Friday's sanctions announcement.
But experts say arranging an inter-Korean summit would be hard after a year in which North Korea conducted its most powerful nuclear test and flight-tested intercontinental ballistic missiles that could, when perfected, target the USA mainland.