Ahead of the opening ceremony, Mr Pence made a brief appearance at a dinner for visiting foreign dignitaries, which was also attended by Kim Yong-nam, the highest-ranking North Korean official to travel to the South.
Saturday's lunch at the Blue House, as Moon's presidential compound is known in Korea, came shortly after Moon and Kim Jong Un's sister enthusiastically cheered on a unified Korean team as they walked around a stadium in Pyeongchang.
Clad in coordinated outfits of red with white and blue accents, North Korea's throng of more than 200 cheerleaders are stealing the spotlight at the 23rd Winter Olympic Games in South Korea as they chant, sway and dance in unison.
Pence shied away from public criticism of Moon when they met Thursday evening, congratulating South Korea on hosting the games and pledging continued support in addressing the North's nuclear threat.
She verbally conveyed Kim's invitation to Moon "to visit the North at his most convenient time", he added.
The pair signed a peace declaration calling to end the armistice with a permanent treaty, but progress stalled and the two sides remain in a stalemate.
The North's presence has dominated the headlines in the early days of the Pyeongchang Winter Olympics. "And that is, demand at the outset of any new dialogue or negotiations that the Kim regime put denuclearization on the table and take concrete steps with the world community to dismantle, permanently and irreversibly, their nuclear and ballistic missile programs".
But the team's formation has provoked a backlash in the South, with critics saying Seoul was depriving its own citizens of the chance to compete on the Olympic stage in its attempts to engage with the North.
Her brother, North Korea's leader has not set foot outside of his homeland or met a single head of state since he assumed power upon the death of his father, Kim Jong-il, in late 2011.
There is a place and time for such a thing but one can not but wonder if this was the occasion given what the North and South Koreans are trying to achieve after year of tensions over Pyongyang's nuclear programmes. Seafood, once a major North Korean export, was now more widely available at home after United Nations sanctions banned these exports in August.
The two Koreas marched together and South Korea's president shared a historic handshake with Kim Jong Un's sister as the Pyeongchang Winter Olympics opened in a spirit of intense rapprochement on Friday.
"The grievances that the world has about North Korea are very legitimate".
The spokesman says Moon also called for a quick resumption of dialogue between the United States and North Korea.
The overture was conveyed by Kim's sister, Yo Jong, who is now in the South to attend the Winter Olympics, according to Deutsche Welle.
North Korean officials, including Kim Jong Un's sister, Kim Yo Jong, were seated one row behind Pence.