"We discussed the need to end the violence and ways and means of doing so. We agreed on an approach which I will share with the armed opposition. I also stressed the importance of moving ahead with a political dialogue which the president accepts," Annan told reporters in Damascus after the meeting.
Annan said Assad reassured him of the Syrian government's commitment to the six-point peace plan, which "we should move ahead to implement in a much better fashion than has been the situation so far," the United Nations said in a release.
"So I am leaving Syria, but we will continue our dialogue and, as I said, the approach we have discussed about ending the violence will also be shared with the armed opposition," said Annan, who brokered the plan in March but it has nothing to stop the violence. "I have a team here on the ground that will continue to do that. I also encourage governments and other entities with influence to have a similar effort."
From Syria, Annan went to Iran to discuss Syria and "to see how we can work together to help settle the situation," the envoy's officials said.
At least 30 people were killed Monday, the Local Coordination Committees of Syria opposition group said. At least 60 people were killed in fighting Sunday, the group said.
The organization said house-to-house raids and arrests were conducted Monday in Daraa and "very intense shelling" was reported in Homs, with helicopters flying overhead.
Syrian state-run TV, meanwhile, aired Sunday what it said were "confessions of four terrorists" who purportedly admitted to committing murder, rape, abduction and robbery, and smuggling weapons and gunmen in Homs, CNN said.
Monday's Annan-Assad talks came after an emergency meeting Annan called in late June of foreign ministers who agreed a transitional governmental body was needed to help end the 16-month Syrian crisis.
Assad has never indicated he would step aside, repeatedly blaming the uprising on home-grown or foreign terrorists bent on destabilizing Syria.
Assad, in an interview that aired Sunday, said the unity government outlined in Annan's latest plan already exists, pointing to local elections in December in which opposition members won a few parliamentary seats.
Assad told German broadcaster ARD he had a two-pronged approach to ending the fighting. The first was to combat the terrorists and the second was "to make dialogue with the different political components ... ."
The United States "is part of the conflict," he said during the interview. "They offered an umbrella -- political support -- to those gangs to create instability, to destabilize Syria."
The opposition Local Coordination Committees of Syria said at least 60 people were killed in fighting Sunday while the state-run Syrian Arab News Agency, reported Sunday authorities thwarted an attempt by armed fighters to enter Syria from Turkey, CNN said.
Reports of violence can't be independently confirmed because Syria severely restricts access of international journalists.
The Syrian military conducted weekend exercises simulating defense of its borders from an invasion, Day News Press said.
"This regime continues to mount despicable attacks against Syrian civilians, and the violence that the regime is perpetrating must end," U.S. Defense Department spokesman George Little said.
While the Pentagon hasn't classified the exercise as "provocative," Little said, that judgment could change based on events.