The evacuation is part of a deal reached on Thursday between the armed rebels and the government army, which allows over 4,000 civilians to be evacuated to displacement shelters in government-controlled areas.
On Friday, US Secretary of State John Kerry said he and his Russian counterpart Sergei Lavrov "have achieved clarity on the path forward" on ceasefire discussions, but there were "narrow issues" still to resolve.
As the first white government bus carrying evacuees emerged from Daraya carrying mostly women and children, Syrian army soldiers swarmed the vehicle, shouting pro-Assad slogans. Some 700 rebel gunmen were evacuated in the dal and are being taken to the rebel-held Idlib Province.
Syrian rebel-groups backed by Turkey say they are clashing with fighters affiliated with Kurdish forces in northern Syria, in a bid to wrestle control of territory from them south of a town they recently seized from Islamic State militants.
The surrender of Darayya marks a success for the government of Syrian President Bashar Assad, removing a persistent threat only a kilometers miles from his seat of power.
It also said a second group of civilians from Daraya left the city by bus.
The agreement is seen as a major defeat for the opposition and has provoked anger among its supporters.
But the two big powers have to persuade the many different groups now fighting, and that is clearly proving hard. "It's hard, but we have no choice", he said.
Daraya-based opposition activist Hussam Ayash said residents were "trying to absorb the shock" of suddenly having to leave.
Daraya is part of "Rural Damascus", a province that includes the capital's suburbs and farmland.
But after being surrounded by loyalist forces for four years and subjected to constant heavy bombardment, life in Daraya had deteriorated to intolerable levels.
"The world is watching", de Mistura said.
An Associated Press journalist who entered the suburb Friday saw a landscape of severely damaged and deserted buildings, some of them charred.
Some 800 gunmen and 2,400 civilians were evacuated, according to the Syrian Observatory for Human Rights.
Activists have accused regime forces of cutting off rebel-held areas from food and water supplies by waging long seiges on rebel-held areas in order to force evacuation deals in what has been dubbed "starve or surrender" tactics.
Located just southwest of Damascus, Darayya has become one of the most heavily-bombed militant-held areas during Syria's conflict, which is now in its sixth year.
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