But Tillerson appeared more conciliatory on Monday.
US Secretary of State Rex Tillerson said the US would be open to dialogue "when the conditions are right".
There was no direct reaction from North Korea to the remarks but in a statement after the USA secretary of state made his comments, Pyongyang responded robustly to the new sanctions by saying it would exact "thousands-fold revenge" on America. He's not specifying what those are.
South Korean Foreign Minister Kang Kyung Wha expressed hope that she could meet Mr Ri.
"Lavrov called on all parties involved to show the utmost restraint in order to avoid the implementation of negative power scenarios on the Korean Peninsula and immediately start searching for a political and diplomatic settlement through a comprehensive solution of the peninsula's problems, including its denuclearization", the statement said.
Tillerson has no plans to sit down with North Korean Foreign Minister Ri Yong Ho at the event.
In their first meeting since Kono took up his new post in last week's Cabinet reshuffle, he and Tillerson agreed to hold the next round of four-way security talks between their foreign and defense ministers "at an early date", according to the Japanese official. "We haven't really discussed that among ourselves", said Philippine acting Foreign Affairs Secretary Enrique Manalo of the USA call to minimize relations with Pyongyang.
Representing the United States at the dinner was Susan Thornton - acting assistant secretary of State for East Asia and Pacific Affairs.
The United States did not seek regime change, the collapse of the regime, an accelerated reunification of the peninsula or an excuse to send the USA military into North Korea, Tillerson said.
South Korea's former foreign minister, Yun Byung-se, also talked briefly with Ri at the same forum a year ago.
But critics say tightened sanctions have done little to stop North Korea's missile and nuclear development.
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China is the North's economic lifeline. One is that if the Communist regime in Pyongyang falls it could undermine the legitimacy of the Chinese Communist Party too.
The North said it will take an "action of justice", without elaborating.
In a statement released Monday to the government-run Korean Central New Agency, the North Korean government criticized the sanctions and vowed revenge. But there were no signs the USA would acquiesce to China's call for a quick return to negotiations.
"We have no reason to believe China will enforce these new sanctions", Kazianis told USA TODAY. Beijing, however, along with countries like Russian Federation, has been reluctant to take more comprehensive, sweeping measures against Pyongyang, especially after the agreement on the new sanctions last Saturday.
Southeast Asian countries are "not happy" with North Korea's missile tests, Cambodia's foreign minister said on Wednesday, calling them a threat to regional security.
Asked how the US can support the Philippine government in its fight against ISIS without endorsing human rights abuses in President Rodrigo Duterte's bloody crackdown on drugs, Tillerson said there was no conflict.
The United Nations Security Council on Saturday imposed sharply increased economic sanctions on North Korea worth one-third of its annual $3 billion exports in an effort to rein in Pyongyang's nuclear and ballistic missile program.
"The two leaders affirmed that North Korea poses a grave and growing direct threat to the United States, South Korea, and Japan, as well as to most countries around the world", the statement said.
China has called for a resumption of talks on the Korean Peninsula and urged North Korea not to provoke the global community after the UN Security Council voted to impose its toughest sanctions yet on Pyongyang.
China, North Korea's lone major ally, has repeatedly said it is committed to enforcing increasingly tough United Nations resolutions on North Korea, though it has also said what it terms "normal" trade and ordinary North Koreans should not be affected.
Trump's initial optimism about China's willingness to help gave way to public exasperation, with Trump saying Chinese President Xi Jinping had "tried" but that it "has not worked out".
After weeks of US frustration over China's reluctance to take action, Trump's strategy of relying on Beijing's help showed early signs of paying off.
Armed with extraordinary new U.N. sanctions, nations are racing to ensure that North Korea's biggest trading partners actually carry them out, an elusive task that has undercut past attempts to strong-arm Pyongyang into abandoning its nuclear weapons.