Kim Yo-jong, North Korean leader Kim Jong-un's sister who identified herself as a special envoy to the South, personally delivered her brother's invitation to Pyongyang to South Korean President Moon Jae-in.
USA officials have been urging South Korea to be cautious in its rapprochement with the North - a point Pence drilled home in private meetings with Moon on Thursday.
The 2018 Winter Olympics, taking place in South Korea, marked a special return for North Korea, which generally bars worldwide travel for its citizens and last participated in the Winter Olympics 8 years ago.
A senior U.S. administration official told reporters that Pence did not discuss North Korea's invitation to Moon when the two leaders watched short track speed skating together.
It would be the first summit in more than a decade between Korean leaders.
According to a tweet by the Washington Post's Tokyo bureau chief, Ms Kim left a note at the palace expressing hope that "Pyongyang and Seoul will become closer in the heart of Koreans" and for "unification and prosperity in the near future". "So the maximum pressure campaign is going to continue and intensify".
The South Korean president requested both that North Korea re-open dialogue with the USA, and that the Koreas hold talks.
"It is absolutely necessary for the North and the United States to engage in talks at an early date", he cited Moon as saying. Pence was scheduled to leave for the U.S. Saturday night.
The North's representative, its highest-level official ever to visit the South, was placed between Olympics chief Thomas Bach and United Nations secretary-general Antonio Guterres.
Of the 22 North Korean athletes taking part in the Games, 12 have joined the unified women's hockey team who play Switzerland in Gangneung on Saturday evening.
Moon and the DPRK delegation shared a view that the two Koreas should keep a good atmosphere of reconciliation and peace on the peninsula while boosting talks, exchange and cooperation between Seoul and Pyongyang.
The shutdown started just before Friday's opening ceremony, where Moon twice shook the hand of Kim Yo Jong, the first member of North Korea's ruling dynasty to venture south since the 1950-53 Korean War.
"When I was on the ice rink I couldn't really see the spectators in the stadium".
The South's Unification Ministry said steps to improve ties would be led by the two Koreas, but with the support of the global community.
"The participants in the opening ceremony warmly encouraged the players of the north and the south who were marching together, holding the flag of the Korean peninsula bearing the Korean map".
The games have turned into an opportunity to cool tensions between the two Koreas, after initial doubt that North Korea would participate at all.