The UN resolution, proposed by France, calls for safe evacuations, "immediate and unconditional" access to humanitarian aid, and protection of medical facilities and personnel.
The resolution stressed "the importance to ensure the voluntary, safe and dignified passage of all civilians from the eastern districts of AIeppo or other areas, under the monitoring of and coordination by the United Nations and other relevant institutions, to a destination of their choice."
The plan also demanded the UN be allowed access to make sure humanitarian aid reaches those in Aleppo who most need it.
He also confirmed that 20,000 civilians have been evacuated in total from the eastern part of the city.
- United Nations adopts monitoring proposal
- At least 65 buses full of evacuees have left eastern Aleppo since the early hours of Monday
- More than 20,000 people have been evacuated, according to Turkish foreign minister
- Among those evacuated was Bana Alabed, 7, who captured the world's attention with her tweets from Aleppo
- Turkish and Russian foreign ministers to meet Tuesday
Also on Monday, UNICEF Regional Director Geert Cappelaere confirmed that 47 children who had been trapped in an orphanage in east Aleppo had been evacuated, "with some in critical condition from injuries and dehydration."
"The evacuation of these orphans, along with thousands of other children from east Aleppo in the past days, is a glimmer of hope amid a grim reality for the children of Syria," he said in a statement.
"Their safe departure is a testimony to the relentless efforts of humanitarians on the ground, working around the clock for children and their families.
"Many vulnerable children -- including other orphans and children separated from their families -- still remain in east Aleppo and need immediate protection," Cappelaere added. "UNICEF reminds all parties of their obligations under international humanitarian law to protect children, wherever they may be."
Evacuees 'have been through hell'
Conditions on the buses are terrible, according to Abdul Kafi Alhamdo, an English teacher in Syria. "We had to stay four people in a seat," he told CNN via text message.
He said children were vomiting due to lack of food and water, and people on the bus were unable to go to the bathroom or take the medicines they needed. Aid workers had advised the group stay silent "to save your lives."
"Save their souls as soon as possible," stressed Abdul Kafi Alhamdo.
Among those evacuated was Bana Alabed, the seven-year-old
who captured the world's attention with her heartbreaking tweets
Casey Harrity, Director of Programs at Mercy Corps
, which is working with refugees being evacuated out of eastern Aleppo, says the people they are welcoming "have been through hell."
"The level of trauma they have experienced is impossible to describe or comprehend," she said. "We are working very hard to ensure they are received with as much dignity and support as possible."
United Nations humanitarian adviser Jan Egeland said on Twitter Sunday that thousands more were still waiting to be evacuated.
"Hopeful operation will proceed smoothly," Robert Mardini, of the International Committee of the Red Cross, said on Twitter.
The UN plan also demanded "that all parties allow complete, immediate, unconditional, safe and unhindered access for the United Nations and its implementing partners, in order to ensure that humanitarian assistance reaches people through the most direct route in order to meet basic needs."
The plan marks a breakthrough for the UN Security Council, which has come under criticism
for failing to find a political solution to Syria's brutal war
that has killed at least 400,000 people. The council has failed to even agree on days-long ceasefires.
Russia, which has supported Assad with airstrikes since September 2015, has used its veto power as a permanent member of the Security Council six times to shoot down UN resolutions on the conflict, while China has vetoed five of those six.
Meanwhile, an official from the Turkish foreign ministry confirmed to CNN that a meeting will take place Tuesday between foreign minister Cavusoglu and his counterparts from Russian and Iran.
"Our minister is heading to Moscow to talk about Aleppo and Syria," said the official.
"This trilateral meeting was planned for 27th but they decided to do it earlier on the 20th.
"While he is visiting Moscow our minister will have a bilateral meeting with Sergey Lavrov about Turkish-Russian relations as well."
The defense ministers of the three countries are also set to hold a meeting the same day.
Winter will be 'a killer'
The Norwegian Refugee Council is assisting people who have fled eastern Aleppo, many of whom slept outdoors as temperatures plunged below zero. The council is hosting them in camps to the north of the city.
"Our major concern at this point is -- not only in Aleppo but across the country, where there are people being displaced by this fighting -- is that they're living in the open, they're exposed to the elements," Thomas White, the organization's Syria Response Director, told CNN's Cyril Vanier. "And it's cold right now, and this winter in northern Syria will be a killer."
For those staying, life is only getting more grim. The government's grip on eastern Aleppo is now so tight there is little left for civilians.
The Syrian regime was accused of continuing its destructive airstrikes this week in violation of earlier ceasefires.
A harrowing video broadcast by ITV's Channel 4 of what is believed to be eastern Aleppo's last hospital shows a child siting on a bed, covered in dust and blood, visibly traumatized, following an airstrike.
A woman, also bloodied and white with dust, wails in pain, saying she lost all her children in a strike in their home as they slept.