The deal to evacuate Barzeh district mirrors similar agreements for opposition-held territory elsewhere in the country, allowing fighters safe passage in exchange for surrender.
It comes days after the start of a deal brokered by regime backers Russia and Iran and rebel supporter Turkey to put in place "de-escalation zones" where the government and opposition will halt hostilities.
"Armed men and some of their families have begun leaving Barzeh on 40 buses heading towards northern Syria," state television said.
The channel did not specify how many people were expected to leave, but the Syrian Observatory for Human Rights monitor said up to 1,500 people -- mostly rebels -- would leave Barzeh today and head to the northwest province of Idlib.
The evacuation deal was struck late Sunday night, and dozens of people had gathered in Barzeh from the morning.
An AFP photographer saw rebel fighters carrying light weapons looking on as children and women in brightly-coloured headscarves pulled shabby suitcases and duffel bags.
Negotiations were ongoing for a similar deal in the district of Qabun, in Damascus's northeast, which forces loyal to President Bashar al-Assad have been shelling heavily for weeks.
Speaking at a press conference in the capital, Foreign Minister Walid Muallem confirmed that the "reconciliation in Barzeh has begun."
"We are working on Qabun and there is the Yarmuk camp, where talks are underway for the evacuation of armed groups," he added.
Assad's government controls all but six districts of Damascus: Barzeh, Qabun, Jobar, Tishrin, Tadamun, and Yarmuk camp.
But the Barzeh deal follows a model of "reconciliation deals" that have been implemented across the country.
The government touts such deals as the best way to end the six-year conflict, which erupted in March 2011 with anti- regime protests and has killed over 320,000 people.
But the opposition says it is forced into them by heavy government bombardment and siege tactics.
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