The invitation was delivered in a meeting between Moon and the North Korean leader's sister, Yo-jong. Moon had replied, "Let's create the environment for that to be able to happen", according to the presidential Blue House.
At the opening ceremonies Friday, Pence sat stone-faced in his seat as Moon and North Korean officials stood together with much of the stadium to applaud their joint team of athletes.
The Winter Olympics have driven a rapid rapprochement on the Korean peninsula, where fears of a new conflict mounted previous year as the nuclear-armed North carried out a series of missile tests and its most powerful atomic blast to date.
Sputnik: North Korean leader reportedly invited South Korean president to Pyongyang.
The North's official Korean Central News Agency also referred to Moon as president.
The two Koreas are technically still at war after a conflict in the 1950s ended with an armistice, rather than a peace treaty. And, you know, President Moon, has appreciated that.
Based on that assurance, Pence was cited as saying he felt confident he could endorse post-Olympic engagement with Pyongyang.
Pence, who is leading the USA delegation to the Olympics, briefly encountered Kim Yong-nam at the Games but the two tried to avoid directly facing each other. United States Vice-President Mike Pence was also present and led the USA delegation.
"There is no daylight between the United States, the Republic of Korea and Japan on the need to continue to isolate North Korea economically and diplomatically until they abandon their nuclear and ballistic missile program", Pence told reporters aboard a flight back to the U.S.
Moon and the delegates agreed to maintain the conciliatory mood inspired by the Olympics and promote inter-Korea dialogue and cooperation, the presidential office said.
The North's delegation to the Games also includes the country's ceremonial head of state as well as hundreds of female cheerleaders.
But the United States and Japan have warned against North Korea's "smile diplomacy", through which Pyongyang is apparently trying to weaken global economic sanctions against it by getting close to South Korea.