North Korea has dismissed an offer of talks from the South as "insincere".
South Korean Foreign Minister Kang Kyung-wha had raised the issue with her North Korean counterpart Ri Yong Ho on the sidelines of a regional forum on Sunday in the Philippines.
North Korean state media later said there could be no negotiations over its controversial nuclear weapons programme while it faces threats from the US.
The KCNA state news agency said the sanctions were a "violent violation of our sovereignty" and vowed to take action.
The unanimous Security Council decision follows repeated missile tests by the North which have escalated tensions on the peninsula.
South Korean media reported that Ms Kang and Mr Ri shook hands in a brief and unarranged meeting at an official dinner event held by the Association of Southeast Asian Nations (Asean) in the Philippine capital, Manila.
A South Korean foreign ministry official confirmed the meeting to the BBC.
South Korean news agency Yonhap reported Ms Kang as saying that Mr Ri's rejection of the talks proposal appeared to be connected with the new sanctions.
"I told him that [the two offers for talks] are an urgent matter that should be carried out immediately with any political agenda put aside and asked him to proactively react," she was quoted as saying.
The foreign minister of China, which is Pyongyang's closest ally, told journalists on Monday: "My feeling is that the North did not entirely reject the positive proposals raised by the South." Wang Yi added that China also supported the South's initiatives.
KCNA insisted on Monday that North Koreas would continue to its nuclear weapons programme and would "not put our self-defensive nuclear deterrent on the negotiating table" while it faces threats from the US.
The US Secretary of State Rex Tillerson is also at the Asean forum, where he spoke about North Korea.
Noting Russia and China's participation in the unanimous vote, he told journalists it was clear there was now "no daylight among the international community" on their desire for North Korea to stop its tests.
Russia and China have previously differed with others on how to handle Pyongyang, but in recent months have joined in calls for North Korea to stop its missile tests while also urging the US and South Korea to halt military drills.
On Monday, US President Donald Trump spoke to his South Korean counterpart Moon Jae-in over the phone about relations on the Korean peninsula.
They agreed that North Korea "poses a grave and growing direct threat" and the two leaders were committed to fully implementing the latest round of UN sanctions, the White House said in a statement.