"The two leaders agreed on more practical and stronger pressure.to make the North Korean regime realise that further provocation will only bring stronger diplomatic isolation and economic pressure leading to a path of collapse", the South's presidential office said in a statement.
It had already stepped up sanctions against North Korea in response to a nuclear bomb test on September 3, imposing a ban on North Korea's textile exports and capping its imports of crude oil.
South Korean President Moon Jae-in wants stronger pressure against its northern neighbor.
A Hwasong-12 medium-range ballistic missile "zoomed to the sky with dazzling flash and big explosion" and "crossed the sky above Hokkaido of Japan along the preset flight track and accurately hit the preset target waters in the Pacific", Xinhua news agency cited the report as saying.
The South Korean leader underscored the importance of enhancing South Korea's own defense capabilities, along with those of the joint forces of South Korea and the United States.
Those arguments were buttressed from within Trump's administration last month when Steven Bannon concurred that America was essentially bluffing about military options, shortly before he was pushed out as the president's strategic adviser.
Nikki Haley says North Korea will be "destroyed" if it continues with what she describes as "reckless" behavior.
In his first general debate speech to the assembly slated for Tuesday, Trump "will urge all states to come together to address grave dangers that threaten us all", such as terrorism and North Korea's nuclear ambitions, according to McMaster. The South Korean president is set to attend a trilateral summit with Trump, and Japanese Prime Minister Shinzo Abe on Thursday.
How has North Korea responded in the five weeks since President Donald Trump threated Pyongyang with "fire and fury?"
The two leaders plan to work with the global community to implement U.N. Security Council resolution 2375, meant to put maximum economic and diplomatic pressure on North Korea following recent missile and nuclear tests.
Trump also took to Twitter to describe his phone conversation with Moon.
"What has to happen is a senior American official or a trusted confidant of President Trump needs to get in the room with a senior North Korean official or confidant of Kim Jong Un".
"After seeing your capabilities and commitment here today, I am more confident than ever that our options in addressing this threat are both effective and overwhelming", Trump said inside an airplane hangar, surrounded by USA airmen and with a looming stealth bomber positioned behind him.