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.
Civilian contact is strictly banned between the two Koreas, which have technically remained at war since the 1950-53 Korean War ended with an armistice instead of a peace treaty.
He has met with some North Korean defectors while there.
Kim Yo Jong told Moon that relations would improve quickly once he meets with her brother.
In Pyongyang, North Korea's official Rodong Sinmun newspaper also reported on the Games bonhomie, publishing photos of its ceremonial head of state clinking glasses with Moon and of the South's leader also shaking hands with a telegenic Kim Yo Jong.
Her compatriot, biathlete Ole Einar Bjorndalen has the overall record with 13 medals, which the 37-year-old could surpass in Pyeongchang, but she said she was not disappointed with silver. U.S. Vice President Mike Pence, sitting a few seats away in the same Opening Ceremony VIP booth, looked stony-faced as the scene unfolded. Pence was seated in a box with a number of South Korea's leaders.
"For our team meetings it is going through to English to South Korean to North Korean".
"The Vice President is grateful that President Moon reaffirmed his strong commitment to the global maximum pressure campaign and for his support for continued sanctions", said Pence's spokeswoman, Alyssa Farah.
Here's what we know about Kim Yo Jong.
North Korea has sent some of the highest level officials ever to the south for these games, the result of diplomacy by Seoul that led to an agreement between the two countries - still formally at war - to participate together in the Winter Games.
The spokesman says Moon replied that the North and South should continue to work to build conditions so that a summit can take place.
In a January interview with Reuters, U.S. President Donald Trump said North Korea is close to having the capability to deliver a nuclear-tipped missile to the United States, and vowed to prevent that.
Whereas the sister of Kim Jong Un, whether she wanted to be or not, [is] complicit in the murderous starvation regime of her brother who suppresses her people to seek nuclear weapons to threaten the entire world.
The official said Pence and Moon did not discuss the North Koreans' invitation to Moon for a summit in Pyongyang, which would set the stage for the first meeting of Korean leaders in 10 years.
The officials toasted with the Korean liquor "soju" and were served buckwheat and potato pancakes, dried pollock soup, grilled halibut and other dishes.
Regardless, both Pence and the North Korean officials can return home and claim they did what they came to Pyeongchang to do.
First rule: in the presence of guests, do not refer to their leader, Kim Jong Un, by name, or certainly do not mention his nuclear and missile programmes.