In his New Year address, leader Kim Jong Un demanded a stop to the exercises.
Critics - among them US Vice President Mike Pence and Japanese Prime Minister Shinzo Abe - warn of a charm offensive by Pyongyang, which mounted a military parade on Thursday in a show of strength.
Kim Jong Un's decision to dispatch his sister to the games indicated he is keen to mend ties with Seoul, while testing the unity among South Korea, the United States and Japan over sanctions imposed on the North over its missile and nuclear programs. As ever, its front page lead story was on the activities of leader Kim Jong Un - in this instance, being given a gift by Zimbabwe's visiting air force commander. Moon's spokesman said the meeting at the Blue House was held in a "friendly" mood.
He said the USA would take whatever "action is necessary to defend our homeland" against the North Korean nuclear threat, according to an interview with NBC.
The first summit was in June 2000 between Kim Dae-jung, a proponent of the so-called Sunshine Policy, and Kim Jong Il.
Kim, a high-ranking official in Pyongyang's dictatorial regime, had been selected by her brother as the North's delegate to the opening ceremony of the Winter Olympics in Pyeongchang on Friday.
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North Korea and South Korea could soon take part in a historic meeting that might help to end the current nuclear standoff. He criticized the Trump administration for straining too hard to signal disgust of Kim Jong Un's government.
Mr. Moon hasn't decided on whether or when to visit the North.
Friday's procession through the Pyeongchang Olympic Stadium was different than past opening ceremonies for a few reasons. Instead of eating with Moon and the other dignitaries, Pence ate with the American athletes.
In its first reports about the Games, North Korea's state-run media slammed U.S. Vice President Mike Pence on Saturday for what it called "shameful" and "snobbish" behavior not in line with the spirit of the Olympics. Those photos quickly went viral.
In Washington, China's top diplomat Yang Jiechi told Trump that China hoped it and the United States could increase their coordination on the North Korea issue.
This item corrects that Kim Yo Jong is the leader's sister.
Tensions between the United States and North Korea have risen sharply over the previous year, as Kim Jong Un has made rapid progress in his nuclear weapons program and has made clear the program is being developed with the American threat in mind. Pyongyang has repeatedly said it needs a deterrent to stop the United States attacking it.
Moon sees the Games as an opportunity to restore regular communication with North Korea and eventually pull it into talks over resolving the worldwide stand-off over its nuclear programme.