The comments came after a top North Korea's ambassador to the UN, Han Tae Song, warned his country is ready to send "more gift packages" to the US.
The leaders' remarks followed North Korea's latest and sixth nuclear test staged Sunday.
Speaking at the Center for Strategic and International Studies, Congressman Stephanie Murphy, said North Korea's nuclear test are a profoundly risky, defiant, and destabilising event.
She urged the 15-member Security Council to impose the "strongest possible" sanctions to deter him and shut down his trading partners.
"Pyongyang's missile and nuclear programme is a crude violation of UN Security Council resolutions, undermines the non-proliferation regime and creates a threat to the security of northeastern Asia", Mr Putin said at a joint news conference.
Diplomats say the UN Security Council could consider banning North Korean textile exports, barring its airline or stopping supplies of oil to the government and military.
Moon had left his offer of talks open as long as the North refrains from further provocations, and some key officials even argued that South Korea should downscale joint military drills with the US and ease sanctions if the North halts provocations.
"I ask Russian Federation to actively cooperate as this time it is inevitable that North Korea's oil supply should be cut at the least", Mr Moon told Mr Putin, according to a read-out from a South Korean official.
Over the past few days, he has spoken multiple times with Japanese Prime Minister Shinzo Abe.
Turnbull said the half-hour phone call was "a very warm" and constructive discussion.
Donald Trump to reach out to global leaders on the North Korean threat issue.
"I think there's considerable internal unrest associated with the cost of the missiles and nuclear programs, and also because of a bad harvest and impending drought".
South Korea's Defence Ministry said the four remaining batteries of the Terminal High Altitude Area Defense (THAAD) system would be deployed on a golf course in the south of the country tomorrow.
Analyzing North Korea: What could happen next?
The second son Jong-chol suffered from an excess of female hormones after a auto accident, Nam said. It was during this trip that they were told about the putative leader's teenage temperament.