At least 350 people reportedly left rebel enclaves late on Sunday, heading towards other rebel-held territory.
Among those to have left is seven-year-old Bana Alabed, who had tweeted about conditions in besieged areas of Aleppo.
A separate evacuation of government-controlled parts of Idlib province, besieged by rebels, started early on Monday.
Thousands more are waiting to leave east Aleppo amid dire conditions.
The UN Security Council is said to have agreed a compromise to allow UN monitoring of the operation. Russia earlier rejected a French-drafted plan to send UN officials to east Aleppo as "a disaster".
The Security Council meeting will start at 09:00 (14:00 GMT) in New York.
Initial efforts to evacuate the last rebel-held enclaves in the city collapsed on Friday, leaving civilians stranded at various points along the route out without access to food or shelter. Bombardment of east Aleppo has left it virtually without medical facilities.
Despite further setbacks on Sunday, buses and ambulances began moving out of the area after nightfall.