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 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 stand up and tell Russia and Assad to stop the carnage," US Ambassador Samantha Power told the assembly ahead of the vote.
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.
Lebanon, Iraq, South Africa, Nigeria and India were among the countries that abstained from the vote on the non-binding text.
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.
(Reopens FGN 8)
Russian envoy to the UN Ambassador Vitaly Churkin had said a day before the vote that the UNGA vote is not very effective and to expect that it is "going to produce some kind of dramatic U-turn in the situation in Syria is unrealistic."
When asked why Russia did not take part in negotiations on the draft resolution, he said there were things that needed to be included in the draft resolution, which is not strong enough on fighting terrorists in Syria.
The Assembly's measure stressed on the need that all parties to the conflict must fully and immediately implement all provisions of various Security Council resolutions concerning the situation in the country, and underscored that all parties must "take all appropriate steps to protect civilians and persons hors de combat, including members of ethnic, religious and confessional communities."
To that end, it noted that "the primary responsibility to protect (Syria's) population lies with the Syrian authorities."
It also expressed deep concerns at presence of terrorist organisations in the country and condemned attacks and violations of human rights and humanitarian law perpetrated by them.
Highlighting that the only sustainable solution to the
It further urged the Security Council to "exercise its responsibility for the maintenance of international peace and security by taking additional measures to address the crisis in the Syria, in particular the devastating humanitarian crisis, and stresses in this regard Article 11 of the UN Charter."
Since the crisis erupted in 2011, the humanitarian situation in the country has taken a downward spiral with more than 13.5 million Syrians now in need to humanitarian assistance and nearly 6.3 million people internally displaced.
The conflict has also killed hundreds of thousands of people, including many children. Almost a million people (974,080) remain trapped in besieged areas and nearly 3.9 million people in hard-to-reach areas.
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