Pence said the US strategy now involves following up inter-Korean talks with US-North Korea talks, and that Moon pushed the North Koreans to talk to the US at the Olympics.
North Korean leader Kim Jong-un has invited South Korea's president Moon Jae-in to a meeting in Pyongyang.
If the inter-Korean meeting at the highest level is to be convened in Pyongyang, it would be the first in a decade following the ones in 2000 and 2007, held in the North's capital during the years of the so-called "Sunshine Policy", which marked the closest relations to date between the two countries.
Any summit between the two still-officially warring Koreas would be a coup for Moon, who has been pushing for a diplomatic solution to the standoff over North Korea's nuclear and missile programmes.
During their meeting President Moon was pictured smiling and shaking hands with Kim Yo Jong.
The official media of North Korea has praised the importance of the historic trip to the South of a delegation that included the sister of the regime's supreme leader, and says the visit paves the way for peace.
He also insisted that Washington was on the same page with its allies, saying "there is no daylight between the United States, the Republic of Korea, and Japan on the need to continue to isolate North Korea economically and diplomatically until they abandon their nuclear ballistic missile program".
Pence did not directly address news of the invitation when asked.
The South Korean leader said in response "let's create the environment for that to be able to happen", Blue House spokesman Kim Eui-kyeom told a news briefing.
"The inter-Korean relations should be mended by the parties concerned at any cost", Moon was quoted by the agency as adding.
North Korea has a track record of reaching out in talks following previous crises on the divided peninsula only to pull out after wresting concessions and aid from the West and analysts cautioned the rapprochement with the South may be short-lived. "But if you want to talk, we'll talk".
Even though the pain inflicted on North Korea by the USA and global community looks to be bearing fruit, few experts project a good outcome. "Moon should be aware that he has to urge denuclearisation to North Korea by himself, not depending on dialogue between the US and North Korea", an editorial in the Yomiuri Newspaper read.
Pence endorsed a post-Olympics engagement after Moon assured him that the North Koreans wouldn't get economic or diplomatic benefits for just talking - only for taking concrete steps toward denuclearization.