Moon also encouraged the North Koreans to be more active in pursuing talks with the United States, saying that "early resumption of dialogue (between the two) is absolutely necessary for developments in the inter-Korean relations as well".
Though South Korea is hosting the Winter Olympic Games for the first time, the nation has some experience in the world of global athletics, having hosted the 1988 Summer Games in Seoul.
On Saturday, he tweeted that the USA would not "allow the propaganda charade by the North Korean regime to go unchallenged on the world stage".
The last summit was held in October 2007 between Roh Moo-hyun and Kim Jong-il, the father of the North Korean leader.
"What the president said exactly is let's create the right circumstances to make it happen", Yoon said in a statement.
But Pyongyang's nuclear programme will hang over any attempts to bring the countries closer together.
The US administration has sought to maintain pressure on North Korea through sanctions and tough rhetoric from President Donald Trump.
Read alsoKoreas to march under single "united" flag in Olympic Games - BBCShe made the invitation during a rare lunch meeting between representatives from the two rival neighbors at Seoul's presidential palace.
Abe, Moon affirm "pressure" stance on N. Korea ahead of Olympics
But his efforts have wider implications for South Korea's foreign policy, most notably its relationships with Japan and the US. The two sides stressed on Saturday that they were interested in maintaining movement toward peace .
Despite the public shows of friendliness, experts have warned the underlying tensions have not gone away.
But the North was welcomed with open arms to what South Korean President Moon Jae-in called "Olympic games of peace".
Amid such mixed feelings, more attention was focused on the preliminary round that featured the combined Korea than any other events, according to Park Jae-yeong, 57-year-old businessman who watched the game on television.
Pence did say, however, that Moon shared with him details of his meeting with North Korean leaders, without elaborating.
North Korea has sent a delegation of 22 athletes to compete in five disciplines, with their women's ice hockey players to compete in a unified Korean team.