Measure drafted by Canada passes by a vote of 122 to 13 in the 193-nation assembly, with 36 abstentions. Russia, Iran and China oppose the resolution.
The UN General Assembly on Friday demanded an immediate ceasefire in Syria and urgent aid deliveries in a resolution adopted by a strong majority as Syrian forces launched another assault on Aleppo.
The measure drafted by Canada passed by a vote of 122 to 13 in the 193-nation assembly, with 36 abstentions. Russia, Iran and China opposed the resolution.
The vote came as Syrian regime leader Bashar al-Assad's forces carried out more air raids on Aleppo and the United Nations reported that hundreds of men had gone missing after fleeing the battered city.
"This is a vote to stand up and tell Russia and Assad to stop the carnage," US Ambassador Samantha Power told the assembly ahead of the vote. "This is a vote to defend the bedrock principles of how states should act, even in war."
Power appealed to Syria and its allies to allow civilians to leave eastern Aleppo and give rebel fighters safe exit from the city. "Do not send them to be tortured in regime prisons," she said.
Russian Ambassador Vitaly Churkin accused the United States of engaging in "aggressive rhetoric" and argued that the resolution had "major shortcomings" by failing to emphasise the need to combat "terrorist groups".
Lebanon, Iraq, South Africa, Nigeria and India were among the countries that abstained from the vote on the non-binding text.
The resolution demands a "complete end to all attacks on civilians" and the immediate cessation of hostilities.
It calls for lifting of all sieges and the "rapid, safe, sustained, unhindered and unconditional humanitarian access" throughout Syria.
Russia and China this week vetoed a draft Security Council resolution calling for a seven-day ceasefire in Aleppo, the Syrian city that is on the verge of falling to regime forces.
It was the sixth time that Moscow, a close Assad ally, has used its veto in the council to block action over Syria.
A crackdown by Syrian regime leader Bashar al-Assad on pro-democracy protesters in 2011 sparked a civil war and Daesh used the chaos to seize territory in Syria and Iraq. Half of Syria's 22 million people have been uprooted and more than 400,000 killed.
Moscow launched an air war in support of Assad's forces last year, while Washington has supported rebel forces battling the regime.