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.
Lebanon, Iraq, South Africa, Nigeria and India were among the countries that abstained from the vote on the non-binding text.
The vote came as Syrian President 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 defend the bedrock principles of how states should act, even in war."
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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 emphasize the need to combat terrorist groups.
It calls for lifting of all sieges and the "rapid, safe, sustained, unhindered and unconditional humanitarian access" throughout Syria.
Canada drafted the text, part of the 193-nation assembly's attempt to break the deadlock over Syria at the Security Council.
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 government forces.
It was the sixth time that Moscow, a close Assad ally, has used its veto in the council to block action over Syria.
Moscow launched an air war in support of Assad's forces last year, while Washington has supported rebel forces battling the regime.
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