The invitation came during talks and a lunch Moon hosted with Kim at the presidential Blue House in Seoul. About 800 conservative protesters gathered in central Seoul to tear up photos of Kim Jong Un and condemn Moon for being too soft on North Korea.
South Korea's president, Moon, welcomed the possibility of a meeting with the North Korean leader, saying the two Koreas should "work together to create the environment to make it happen", a spokesman said.
Analysts said his acceptance was assured.
"Moon should make it clear that he can have an inter-Korean summit only when the North promises to abandon its nuclear program", she stressed. It will be an extraordinary moment for Koreans, and it comes just 24 hours after the opening ceremony.
As he met with Pence on Thursday, Moon added his hope that it becomes "a venue that leads to dialogue for the denuclearization of the Korean Peninsula".
The United States insisted it would continue its policy of exerting worldwide pressure on North Korea in an effort to get it to abandon its nuclear program.
It cemented what has been a rapid improvement in Korean ties since North Korea - after months of fierce nuclear rhetoric and missile tests - agreed last month to attend its first Olympics in the South.
"We enjoyed sea breeze, laughed and had some delicious sodas", goaltender Shin So Jung said in Korean.
The Trump administration is cautious of Seoul falling for North Korea's charm offensive during the Winter Olympics, which are taking place amid tension over Pyongyang's nuclear programme.
Expectations are sky-high for an array of stars at Pyeongchang, including American skiers Mikaela Shiffrin and Lindsey Vonn, while the drama in figure skating centres on whether Japan's "Ice Prince" Yuzuru Hanyu can recover from injury to retain his crown.
Washington, meanwhile, "looks like the bad guy to some, and has no plan for how to prevent alliance decoupling", he added.
President Donald Trump and the North's leader Kim Jong Un have engaged in a series of personal insults.
Kim Jong Un invites South Korean president for summit: South Korea
Regardless, both Pence and the North Korean officials can return home and claim they did what they came to Pyeongchang to do. Before now, she had been seen mainly in photos, but often in the background, as her brother took center stage.
But amid the thaw on between the two Koreas, and the ensuing calming of tensions on the peninsula, Washington's has taken a hit, appearing nearly obstinate.
On Saturday North Korean leader Kim Jong-un invited the South's President, Moon Jae-in, to Pyongyang for talks.
The shadow of a nuclear war may still hang over the Winter Olympics in South Korea this week, but Pyongyang's decision to send its athletes rather than sappers and saboteurs has calmed fears about a violent disruption.
Many experts say Kim's overture is meant to use improved ties with Seoul as a way to weaken USA -led global sanctions toughened after its series of big weapons tests past year that include its sixth and largest nuclear test explosion and three intercontinental ballistic missile launches.
North Korea has spent years developing its military, saying it needs to protect itself from USA aggression.
Pence did say, however, that Moon shared with him details of his meeting with North Korean leaders, without elaborating.
They were cheered wildly by a crowd which appeared loudest for the North Korean performance, although none of the spectators wore national symbols representing either side.
The first Kim dynasty member to visit the south since the 1950-53 Korean War, Kim Yo Jong sat alongside South Korean President Moon Jae-in to watch the Koreans suffer an 8-0 shutout by Switzerland in Pyeongchang.
Washington calls it a campaign of "maximum pressure".
That said, these officials said that if the North Koreans had approached Pence with pleasantries, he would have responded in kind.
"The North Koreans should understand that for a summit or any kind of serious talks to occur, Moon needs to be able to take something to Washington - something that addresses denuclearisation", he said before the North's invitation to Moon was announced.