It passed a non-binding resolution, which also condemns the Syrian government’s use of heavy weapons, by 133 votes to 12 with 31 abstentions.
The move came after the resignation of UN envoy Kofi Annan and failure of his six-point peace plan.
Government forces backed by tanks have launched a new assault in Damascus.
Shelling also continued on Friday in Syria’s largest city, Aleppo.
Activists say more than 20,000 people – mostly civilians – have died in 17 months of unrest.
The resolution passed at the UN expresses “grave concern” at the escalation of violence in Syria and deplores “the failure of the Security Council to agree on measures to ensure the compliance of Syrian authorities with its decisions”.
“The first step in the cessation of violence has to be made by the Syrian authorities,” the resolution said.
Abdallah Al-Mouallimi, the envoy for Saudi Arabia which is the driving force behind the resolution, had urged the Assembly to maintain its moral and humanitarian values by approving the resolution.
Syria’s envoy, Bashar Jaafari, reacted to the passing of the resolution by saying his government still supported Kofi Annan’s six-point plan.
Accusing Saudi Arabia and Qatar of having undermined the plan before coming out in support of it, he said: “You cannot be a fireman and an arsonist at the same time.”
UN Secretary General Ban Ki-moon said the conflict in Syria had become a “proxy war” and called on powers to overcome their rivalries in an effort to end the violence.
“The acts of brutality that are being reported may constitute crimes against humanity or war crimes,” he said.
Russia and China have blocked attempts in the UN Security Council to impose sanctions against Damascus.
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