Trump announced the suspension last week after meeting with North Korean leader Kim Jong-un in Singapore, saying it would be "inappropriate" to continue the "provocative" drills while negotiations to dismantle the regime's nuclear weapons program are under way.
"South Korea probably knows North Korea better than anybody" due to proximity and "all that painful history", she said, adding: "We will bring all that experience and lessons learned to the table, to move the agenda forward".
The two-day training - tiny compared with the suspended US-South Korea war games - will involve six warships and seven aircraft while a unit of marines will land on the largely bare rocky islets, inhabited by around 40 people - mostly police officers.
Seoul, which has tens of thousands of U.S. troops on its soil to help protect it from its hostile northern neighbor, said the suspension would affect the large-scale Ulchi Freedom Guardian exercises slated for August. "I think what is left now is how completely and swiftly we implement such a wonderful agreement", he said.
Diplomatic efforts have gathered pace since then, leading to a landmark summit between Kim and US President Donald Trump in Singapore last week.
Trump also said that halting the drills is reasonable as they are "tremendously expensive" and "provocative" in an announcement that took Washington's security allies in Asia and some USA defense officials and experts by surprise.More news: China's Xi praises North Korea's Kim for Trump summit, promises support
US soldiers and vehicles prepare to participate in the Ulchi Freedom Guardian near the inter-Korean border in Paju, north of Seoul on August 22, 2016.
It means this year's exercise has been postponed indefinitely.
The U.S. -South Korean exercise calendar hits a high point every spring with the Foal Eagle and Max Thunder drills, which both wrapped up last month.
The two Koreas held talks on sports exchanges on Monday in the border village of Panmunjom in the demilitarized zone, but World Cup coverage in the North was not on the agenda, according to officials at South Korea's unification ministry. A meeting at the Pentagon between the Secretary of Defence, Secretary of State and the National Security Advisor has been convened for later in the week to discuss the issue, the statement said.
The athletes from the two sides will march together under a unified peninsula flag and the country name "Korea" during the opening and closing ceremony of the Asian Games, as they did in the Winter Olympics in South Korea in February.
The Ulchi Freedom Guardian drills have taken place annually for more than 40 years.