The event, cheered by the crowd at the Gangwon Career Education Institute in Sokcho, featured teams from both countries performing for 25 minutes before briefly joining together as one.
"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 ballistic missile program", Pence told reporters on the flight back to the US.
"Let us promote mutual conciliation and harmony through the power of sport".
The North Korean delegation to the Olympics proposed the first-ever meeting between North Korea's shadowy leader another head of state, to take place with Moon "at an early date".
The officials, who spoke on condition because they were not authorized to discuss the USA approach publicly, also denied that Pence had been blindsided by the seating arrangement - with the North Koreans in the row behind him, allowing Kim Yo Jong to be easily pictured in profile next to the vice-president.
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 previous year that include its sixth and largest nuclear test explosion and three intercontinental ballistic missile launches.
Hundreds of Twitter users in recent days have referred to "Pyongyang2018", and while some were making a dig at the presence of North Korea at the Games, in many cases it appears to have been a simple mix-up.
Fans unfurled a large banner reading "We are one" after the game and then IOC President Thomas Bach and South Korean President Moon Jae-in joined the powerful sister of North Korea leader Kim Jong Un, Kim Yo Jong, and Kim Yong Nam, North Korea's nominal head of state, for photos with the team.
On his flight to Alaska on Saturday, the vice president said he left Asia "encouraged that we will continue to work very closely to continue and intensify the maximum pressure campaign" against North Korea.
North Korea's delegation to the games will conclude its visit later on Sunday, having appeared to thaw the rift with the South, but there is scepticism about how honest Pyongyang is about improving relations.
Moon warmly shook hands with her and later held talks with her in Seoul.
Tensions soared previous year as the North staged a series of nuclear and missile tests in violation of United Nations resolutions, while leader Kim and United States President Donald Trump traded colorful insults and threats of war.
A North Korean art troupe returned home on Monday, ending a high-profile performance tour of South Korea to celebrate the Pyeongchang Winter Olympics.
South Korean officials have said Moon was invited to Pyongyang to speak with Kim Jong Un during the talks and lunch that Moon hosted at the presidential Blue House in Seoul on Saturday.
Though it has no real medal contenders and sent only 22 athletes, North Korea has turned out to be a major political player in Pyeongchang.
"These actions cannot replicate the reality of our dear brothers and sisters, but with prayerful reflection they can provide a window into the world of a North Korean Christian".
Vice President Mike Pence's efforts to keep North Korea from stealing the show at the Winter Olympics proved short-lived, drowned out by images of the two Koreas marching and competing as one.
The statement from the Ministry of Unification came after the North Korean delegation concluded its three-day visit.
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