The evacuation began days after regime backers Russia and Iran and rebel supporter Turkey signed a deal to implement "de-escalation zones" where the government and opposition will halt hostilities.
Foreign Minister Walid Muallem on Monday rejected any role for United Nations or international forces in monitoring the zones.
The deal to evacuate Barzeh district mirrors similar agreements for opposition-held territory elsewhere, allowing fighters safe passage in exchange for surrender.
"Armed men and some of their families have begun leaving Barzeh on 40 buses heading towards northern Syria," state television said.
It added that the evacuation would continue for five days, but that rebel fighters who chose to stay could do so if they register with the government.
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 gathered in Barzeh from Monday 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.
Fighting continues despite de-escalation deal
A source from the pro-government National Defence Forces said rebel fighters would be allowed to take their "personal weapons" with them.
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.
"We are working on Qabun and there is the Yarmuk camp, where talks are underway for the evacuation of armed groups," Muallem said during a news conference Monday.
'De-escalation' zones
Assad's government controls all but six districts of Damascus: Barzeh, Qabun, Jobar, Tishreen, Tadamun, and Yarmuk camp.
The evacuation is a first in the capital, but the Barzeh agreement 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.
Monday's evacuation began as fighting subsided across the country after a deal signed by Russia, Iran and Turkey to implement four "de-escalation zones".
The agreement, inked in the Kazakh capital Astana, does not include Damascus city.
It mandates an end to hostilities, including air strikes, in specified zones for six months.
The deal came into effect on midnight Friday, with fighting subsiding, but the co-sponsors have until June 4 to submit maps finalising the borders of the zones.
It calls for continued fighting against the Islamic State group, which is largely absent from the areas where the zones will be implemented, as well as former Al-Qaeda affiliate Fateh al-Sham Front.
Fateh al-Sham has a powerful presence particularly in Idlib province, which is included in the zones, where it is allied with other opposition groups.
The deal says the "de-escalation zones" would be bordered by security zones with checkpoints and observation centres "ensured by the forces of the guarantors by consensus", but that "third-party" monitors could also be deployed.
But on Monday, Muallem rejected UN or international monitors.
"We do not accept a role for the United Nations or international forces to monitor the agreement," he said.
He added that there could be a role "as the Russian guarantor has said, for military police", but it was unclear if he was referring to Syrian or foreign units.
Share
