The Trump administration is wary of engagement with the North, which has been subjected to increasingly tough global sanctions, unless it shows clear signs of giving up its nuclear weapons program.
"I am very confident, as President Trump is, that President Moon will continue to stand strongly with us in our extreme-pressure campaign", Pence told NBC in an interview on Friday, maintaining all options were open to deal with the crisis.
Earlier, it has suggested that talks could start between the U.S. and North Korea once South Korea suspends its joint military exercise with South Korea.
"We are not going to comment on the issue".
The Games in South Korea are due to finish on Feb 25.
It is now unclear how the unidentified Korean man was able to jump the barriers and get onto the ice chute but local police in Pyeongchang, the South Korean resort where the two week tournament is held, say they are investigating.
Pence left South Korea Saturday night, flying out of Pyeongchang after attending a short-track speedskating event with Moon - who then went to watch ice hockey along with Kim and the North Korean delegation. The United States fought with South Korea and maintains tens of thousands of troops and an "ironclad" agreement to protect its ally.
While it's unclear whether the displays of unity will reduce tensions between the two, with North Korea testing several missiles and making world headlines for its provocative acts in the past few months, we can still dream.
Kim Yo Jong is the first member of her family to set foot in this country since the north and south were divided and her grandfather, Kim Il Sung, founded North Korea. The supporters waved blue and white flags that showed an undivided Korean Peninsula.