Both leaders agreed to explore "a new, creative and out-of-the-box approach" to resolve the issue "in our generation".
Aso said: "By making progress in the negotiations over the biggest issue that lies between the two nations - the final resolution of the territorial issues - I strongly hope for the building of Japan-Russia relations that are befitting of true partners in this region."
The prime minister stressed Japan's interest in forging "strategic ties" with Russia and said "we regard Russia as an important partner in the Asia-Pacific region".
Japanese officials, however, said that Tokyo's basic position on the dispute would remain the same.
Without directly referring to the Kuril Islands dispute, Medvedev said: "I welcome the fact that our relations are developing on different levels.
"We met in Lima, we are meeting on Sakhalin, we are soon going to London, and then we will meet for the G8 in Italy."
The president was referring to last year's Asia-Pacific Economic Co-operation summit in the Peruvian capital, the forthcoming meeting of the G20 summit in the UK capital in April and the summit of the G8 leading economies in July.
Medvedev said: "Political contacts have taken on a good intensity. Trade between Russia and Japan is stable, is growing and for the year 2008 totalled $30bn. This is a good figure."
"I am glad to have the opportunity to meet with you and discuss our bilateral ties, all the more so that this linked with the opening of our LNG plant".
He described the launching of the plant "as a sign that we are prepared to work with our partners in the most different situations, including complicated ones [the global financial crisis]".
The plant on Sakhalin Island, located about 150km from Japan's northern island of Hokkaido, will supply gas to Japan as it seeks to diversify its energy supplies.
Russia ceded half of Sakhalin to Japan after the war between the countries in 1905, with Moscow taking back the whole of the island in 1945.