Song and Aso held their first meeting since Song's appointment was announced earlier this month, on the sidelines of this year's Asia-Pacific Economic Co-operation forum.
"The two sides agreed on the need to make North Korea abandon its nuclear weapons possession," a Japanese official said.
"They confirmed that it is necessary to use both pressure and dialogue firmly," the official said, adding that the two men affirmed their close ties with the United States over the issue.
Song and Aso also agreed to make an effort to produce "concrete results" at planned six-way talks on ending the North's nuclear program, the official said.
After conducting its first atom bomb test last month, North Korea agreed to return to the six-nation disarmament talks, which will draw delegations from South and North Korea as well as China, Japan, Russia and the United States.
During the meeting, Song agreed to visit Japan "at the earliest timing", the official said.
South Korean President Roh Moo-Hyun named Song on November 1 as the successor to Ban Ki-Moon, the next United Nations secretary general, who succeeds Kofi Annan in January.
Song, a 58-year-old career diplomat, was previously South Korea's lead delegate to the six-nation talks.

Copyright AFP (Agence France-Presse), 2006