The spokesman said Mr Moon also called for a quick resumption of dialogue between the United States and North Korea.
The invitation was delivered in a meeting between Moon and the North Korean leader's sister, Kim Yo Jong.
The team had included 12 North Korean players. White House officials stressed that Pence had applauded only for the American team, but Asia experts said the vice president's refusal to stand could be seen as disrespectful to the hosts.
Mattis cited a recent meeting with South Korea's Defense Minister where they both agreed that the US and South Korea are united against North Korea. The crowd at Kwandong Hockey Centre came alive each time Korea took possession, as did North Korea's all-female cheerleading squad.
The presence of the delegation at the Games is the result of agreements signed by the two Koreas, who have technically been at war for 65 years.
"We're getting along great", the Kim Jong-un lookalike told Reuters.
Kim makes halfpipe final with top mark, Clark just advances
Official rules stated that competitors for snowboard halfpipe must have turned at least 15 years old by the end of 2013. Teammates Arielle Gold, Maddie Mastro and Kelly Clark went second, third and fourth behind Kim at X Games.
Accepting North Korea's demand to transport more than 100 members of the art troupe by sea, South Korea treated the Mangyongbong-9 ferry as an exemption to the maritime sanctions it imposed on the North.
"I would love the team to get the Nobel Peace Prize", Ruggiero, a member of the IOC's executive board said a day after the unified Korean team competed at the Pyeongchang Olympics.
USA student Otto Warmbier cries at court in an undisclosed location in North Korea, in a photo released by North Korea's Korean Central News Agency (KCNA) in Pyongyang on March 16, 2016.
"Let's not whitewash North Korea's record".
Pence did say, however, that Moon shared with him details of his meeting with North Korean leaders, without elaborating.
While political tensions seem to be on the rise more than ever before, it's the athletes who are sharing their time in the spotlight to help pave the way to better communication, and, in the hopes of many Koreans, eventual reunification.