Pyongyang fired an intermediate-range Hwasong-12 over Japan early on Tuesday, which it said was a mere "curtain-raiser" for the North's "resolute countermeasures" against ongoing U.S.
Trump and Moon during their phone conversation also agreed, according to the South Korea's Blue House, to enhance Seoul's capability of deterring the North Korean threat by giving it missiles with greater capabilities.
Reports of the event came hours after North Korea claimed it had developed a hydrogen bomb that could be mounted on the its new intercontinental ballistic missiles.
Despite Pyongyang's snub of Moon's invitations for talks in July, the South Korean president continued to emphasize the importance of dialogue with its neighbor in the phone call with his American counterpart.
Moon is also pursuing a diplomatic route to contain the threat from North Korea. Japan also raised the prospect of further sanctions, saying curbs on North Korea's oil trade would be on the table.
Trump had also given his "conceptual approval" for the planned sale of billions of dollars of USA military equipment to South Korea, the White House said.
Trump made his remarks to reporters while visiting hurricane-hit Houston a day after he spoke with South Korean President Moon Jae-in and struck a deal allowing Seoul access to longer-range missiles as well as a potential arms sale.
That followed the test launch of two long-range ballistic missiles in July in a sharply lofted trajectory that demonstrated a potential range of 10,000 km (6,000 miles) or more that would put many parts of the USA mainland within striking distance.
"North Korea must realize it can not ensure the safety and development of its regime through nuclear weapons and missiles". Seoul's National Intelligence Service denied the claim.
Earthquakes triggered by North Korean nuclear tests have gradually increased in magnitude since Pyongyang's first test in 2006, indicating the isolated country is steadily increasing the destructive power of its nuclear technology.
A USA official who studies North Korea's military and politics said it was too early to determine if a test supported the North's claim that it has succeeded in developing a thermonuclear weapon, "much less one that could be mounted on an ICBM and re-enter Earth's atmosphere without burning up".
The US now has around 28,500 soldiers stationed in South Korea, and Washington also guarantees the country's protection under its nuclear umbrella.
"The president ordered (the military) to prepare active measures to counter North Korea's reckless provocations based on the strong Korea-U.S. alliance and to maintain complete readiness against additional provocations", Chung said.
A hydrogen bomb can achieve thousands of kilotons of explosive yield - massively more powerful than some 10 to 15 kilotons that North Korea's last nuclear test in September was estimated to have produced, similar to the bomb dropped on Hiroshima, Japan, in 1945.
North Korea escalated regional tensions once again this week by firing a ballistic missile over Japan in defiance of United Nations resolutions and fresh sanctions imposed last month which are expected to slash its export revenues by a third.
Impoverished North Korea and the rich, democratic South are technically still at war because their 1950-53 conflict ended in a truce, not a peace treaty.
The idea of potentially withdrawing seems to have been prompted by the breakdown in negotiations between South Korean officials and the United States trade representative, Robert Lighthizer, an American official with knowledge of the situation said.