The leaders met on the sidelines of the Asia Pacific Economic Cooperation (APEC) summit in the Vietnamese resort city of Danang.
Ties between China and Japan, the world's second- and third-largest economies, have been plagued by a long-running territorial dispute over a cluster of East China Sea islets.
"At the end of the meeting, President Xi said this is a meeting that marks a fresh start of relations between Japan and China. I totally feel the same way," Abe told reporters.
Abe said he has proposed visiting China at an appropriate time, which would then be followed by a Xi visit to Japan.
At the meeting, the two countries agreed to deepen their cooperation on North Korea and to hold a trilateral summit with South Korea at the earliest possible date.
"With the North Korea situation at an important phase, the role China ought to play is very big," Abe said.
China and Japan have also agreed to accelerate talks for an early implementation of a communication mechanism between their military forces, Abe said. He also proposed that Japan and China cooperate in doing business in third countries.
China's statement about the meeting, released by the official Xinhua news agency, cited Xi as telling Abe that stable relations were in both sides' interests, and that they must make unremitting efforts to continue improving ties.
The two countries "must take constructive steps to appropriately manage and control disputes that exist between the two countries", Xi added.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)