In a text message sent to reporters, presidential chief press secretary Yoon Young-chan said the leaders of the two Koreas "candidly exchanged their opinions to implement the joint declaration of the first summit and to successfully hold the US-North Korea summit set for June 12 in Singapore".
Mr Trump had rattled the region on Thursday by cancelling his meeting with Mr Kim, citing "open hostility" from the Pyongyang regime.
The president has also reportedly dismissed concerns by top White House staff and advisers that North Korea was manipulating him, according to The Times.
Saturday's Korean summit came hours after South Korea expressed relief over revived talks for a Trump-Kim meeting.
In a tweet, Trump the summit was "likely" to happen as scheduled June 12.
Trump said Saturday that conversations about a potential summit were "going along very well".
Trump responded to reporters' questions about the summit saying that it was "moving along very nicely".
In addition to those talks, White House spokeswoman Sarah Sanders said a "pre-advance team" left for Singapore - where the summit has been expected to take place - on Sunday morning to work on logistics.
Relations between the two Koreas had chilled in recent weeks, with North Korea cancelling a high-level meeting with Seoul over South Korea's participation in regular military exercises with the United States and insisting that it will not return to talks unless its grievances are resolved.
Moon acknowledged Pyongyang and Washington may have differing expectations of what denuclearization means and he urged both sides to hold working-level talks to resolve their differences. The North has previously used the term to demand the United States pull out its 28 500 troops in South Korea and withdraw its so-called "nuclear umbrella" security commitment to South Korea and Japan.
Kim and Trump's initial decision to meet followed months of war threats and insults between the leaders over the North's nuclear program.
The weekend talks were the latest twist in a week of diplomatic ups and downs over the prospects for an unprecedented U.S.
A statement from North Korea's state news agency KCNA confirmed what the South Korean president said.
Pictures from the summit showed Mr Moon and Mr Kim shaking hands as well as embracing, and revealed Mr Kim brought along his sister, Kim Yo Jung.
"A meeting was held because officials of both countries thought that meeting face-to-face would be better than a phone call", Moon said, noting that the summit was in line with previous agreements for the two leaders to meet more often.