The invitation, delivered by Kim's visiting younger sister, Kim Yo Jong, said Kim was willing to meet the South's leader "at the earliest date possible", a spokesman for the presidential Blue House said.
Images from the lunch meeting at the Blue House - broadcast live on South Korean TV - showed Moon sitting in front of Kim, rather than Kim Yong Nam, North Korea's ceremonial head of state who was technically the more senior official at the meeting.
Saturday's meeting, the most significant diplomatic encounter between the two sides in more than a generation, could now be surpassed should Moon accept Kim's invitation to visit Pyongyang later this year, CNN reported.
The letter delivered by "Special envoy Kim Yo-jong" stated her brother's "wish to improve inter-Korean relations", Moon's spokesman Kim Eui-kyeom said.
Switzerland drew first blood halfway through the first period on a short-handed play and continued to place pucks in the net, their players fist-pumping and cheering as they stretched the goal gap wider and wider.
Pence, meanwhile, had only one event on his schedule on Saturday: he appeared back at Pyeongchang to view the short-track speedskating event, along with second lady Karen Pence and the USA delegation.
Choi and other volunteer helpers at the Games were urged to avoid the topic that another Olympics guest, U.S. Vice President Mike Pence, has been eager to address in his public remarks ahead of the opening ceremony: North Korea's arms programme.
This has exposed the gaps between Seoul and Washington in how to deal with North Korea.
While a summit in Pyongyang would signal warming ties on the peninsula, it also risks driving a wedge in the alliance between the US and South Korea.
After the threats, missile launches and general nastiness from the North over the past year, there is still a lot of anger, fear and scepticism in the South.
The hardline US policy is at odds with the more conciliatory approach towards the North favoured by South Korea. These officials said that while Pence did not greet Kim Yong Nam, he didn't deliberately skip over him.
"A strike also would not stem the threat of proliferation but rather exacerbate it, turning what might be a North Korean moneymaking endeavour into a vengeful effort meant to equip other bad actors against us". The meeting would be the first official talks between the two in 10 years.
The North Korean delegation then took the bullet train to Gangneung, the venue of all ice competitions, and attended a banquet hosted by the South's unification minister Cho Myong-gyon.
As if to drive the point home, Kim Jong Un used the eve of the Olympics to throw a massive military parade in Pyongyang that was highlighted by several huge intercontinental ballistic missiles rolled out in launcher trucks. Kim sent his sister Kim Yo Jong and Vice President Pence was also in attendance as part of the USA delegation. Now the joint women's hockey team is playing their first game, offering a look at how sports diplomacy could work on the Korean Peninsula.
North Korea's nominal head of state Kim Yong Nam also partook in the historic luncheon with Moon.
However, the result mattered less than the political symbolism as around 200 of North Korea's famed "army of beauties" charmed a crowd of 3,600 besotted locals with nostalgic oldies - and even broke out a Mexican wave.
"The grievances that the world has about North Korea are very legitimate".
"The chemistry on the team is better than I ever predicted", said Korea coach Sarah Murray, brushing off the heavy defeat.