Moon, meanwhile, welcomed "America's return" to the world stage and said both leaders pledged to work closely toward denuclearization of the Korean peninsula.
Moon, whose single five-year term is to end next May, said he’ll focus on establishing lasting peace on the Korean Peninsula during his final year in office.
Kim and Moon have decided to opt for peace, with North's promise of shutting down main missile testing sites.
Kim may see the sit-down with Trump as necessary to easing pressure from crushing sanctions and to winning security assurances in a region surrounded by enemies.
North Korea's Kim Jong Un hopes Donald Trump summit will 'end history of confrontation': Moon Jae-in
Donald Trump rattled the region on Thursday by cancelling his meeting with Kim, citing open hostility from Pyongyang.
Trump's cancellation of the summit blindsided treaty ally South Korea, which had brokered the remarkable detente between Washington and Pyongyang.
US national security adviser John Bolton reacted coolly to word that Kim would abandon his weapons if the United States pledged not to invade.
The summit took place at Panmunjom, a small village located in the the demilitarised zone separating the two Koreas.
Kim is the first North Korean leader to set foot on the South's territory since the 1950-53 Korean War ended in a ceasefire.
Kim's news agency said that the leader would 'open-heartedly' discuss with Moon "all the issues arising in improving inter-Korean relations and achieving peace.
At Kim's invitation the two men briefly crossed hand-in-hand into the North before walking to the Peace House building on the southern side of the truce village of Panmunjom for the summit.
North Korea has justified its nuclear strategy citing the presence of nearly 30,000 US military personnel in South Korea.
Seoul said no agreement had yet been reached on a meeting between Kim and Moon, which would be the first such summit since 2007.
Pyongyang carried out its sixth, and largest, nuclear test in September and today's launch was the third successful test of an ICBM seen as capable of reaching much of the US mainland.
South Korea is rolling out the red carpet for Trump as it seeks messages of assurance about the alliance and US resolve.