Both praised the South Korean-U.S. coordination in response to North Korean missile threats in recent days, Gesecki said.
The South Korean military has raised its alert and its nuclear crisis response team into action, according to Korean news agency Yonhap.
Trump has also spoken twice this week with Japanese Prime Minister Shinzo Abe.
U.S. Defense Secretary Jim Mattis told reporters Thursday that the Pentagon was working with Secretary of State Rex Tillerson, who is taking the lead, concerning North Korea.
To protect South Korean citizens and US forces, she added, they discussed the alliance's decision to deploy the Terminal High Altitude Area Defense system, which she emphasized is a purely defensive capability.
The South Korean statement on the call mentions that Trump and Moon "reaffirmed their view that it was important to have North Korea come out to the dialogue table to peacefully resolve the North Korean nuclear issue by applying maximum sanctions and pressure on the North". -South Korean military drills.
Under third-generation leader Kim, North Korea has been pursuing a nuclear device small and light enough to fit on a long-range ballistic missile, without affecting its range and making it capable of surviving re-entry.
The test comes amid heightened regional tension following Pyongyang's two tests of ICBMs in July that potentially could fly about 10,000 km (6,200 miles), putting many parts of the USA mainland within range.
It also objected to the translated title of the South Korean edition as "Capitalist People's Republic of Korea" and the book's cover that replaced the red star in North Korea's official seal with the USA dollar mark.
"The power is 10 or 20 times or even more than previous ones", said Kune Y. Suh, a nuclear engineering professor at Seoul National University.
And repeated rounds of United Nations resolutions and sanctions against North Korea appear to have had little or no impact.
"North Korea's mission is quite clear when it comes to this latest atomic test: to develop a nuclear arsenal that can strike all of Asia and the USA homeland", Harry Kazianis, director of defence studies at the conservative Center for the National Interest in Washington, said.
Tensions on the Korean Peninsula are at their highest point in years after a series of missile tests by Pyongyang. The two countries technically remain at war since a truce agreed to end the 1950-1953 Korean War was never followed by a peace agreement.
"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.
"He reaffirmed the ironclad US commitment to the Mutual Defense Treaty, which states that a North Korean attack on one is an attack on both nations", she said.