Moon, who played a key role in mending relations between Washington and Pyongyang after Trump abruptly canceled the June 12 summit on May 24, said he hopes the summit will open a "new era."
A meeting Moon was to hold with members of his Cabinet, including Prime Minister Lee Nak-yeon, was postponed for about 12 minutes because of the live broadcasts.
Moon watched the first encounter between Kim and Trump from the presidential Blue House and said the South Korean people are engrossed with the events in Singapore.
"I hope it will be a successful meeting that gives us a new era of denuclearization and peace, [a new era] for relations among South Korea, North Korea and the United States," Moon said Tuesday morning local time, according to Yonhap.
In Tokyo, Japanese Prime Minister Shinzo Abe also expressed support over the summit, saying the joint statement reflected Kim's willingness to seek the denuclearization of the Korean Peninsula.
Abe also vowed to expand cooperation with the international community, including Washington, Beijing, Seoul, and Moscow, to settle various agendas surrounding North Korea, especially the return of Japanese abductees in Pyongyang.
"I highly praise and appreciate President Trump for mentioning the issue of abduction [of Japanese by North Korea] to Kim, which is an important issue for Japan," Abe told reporters.
Abe also said Tokyo will directly negotiate with Pyongyang to have Japanese abductees return home.
Geng Shuang, a spokesman of China's Foreign Ministry, said during a regular press briefing that Beijing wishes to play a constructive role to help implement the summit agreement.
"It is very important and positively meaningful that the leaders of the two countries held dialogue on an equal footing," Geng said. "New history has been made and China welcomes and supports it. This is also a goal that contains China's expectations and efforts."
He then expressed the wish that North Korea and the United States would remove obstacles, build mutual confidence and take substantial steps to attain denuclearization and peace on the Korean Peninsula.
"We wish all related countries join the efforts, and China will play its own constructive role," the spokesman said.
Russia also expressed hope over the potential revitalization of six-party talks.
Speaking to Russia's TASS, Deputy Foreign Minister Sergei Ryabkov said he "welcomes the fact that an important step forward has been made."
Russia is willing to "make political, intellectual, practical and creative contributions" to various issues surrounding the Korean Peninsula, he said.
South Korean news agency Yonhap contributed to this report.