In the resolution, the assembly "strongly condemns the continued escalation in the use by the Syrian authorities of heavy weapons" and "ballistic missiles" against civilians.
But references in the text to the opposition Syrian National Coalition infuriated Russia and Assad's UN envoy.
The assembly demanded all sides "rapidly" implement a communique agreed by the major powers in Geneva last year laying out the steps toward a transitional government.
The resolution welcomed the Syrian National Coalition "as effective representative interlocutors needed for a transition." Russia said this would encourage opposition "armed actions" against the Assad government.
Russia and the United States have agreed to press for a new international conference on Syria which is expected to be held in Geneva next month.
But Russia's UN ambassador Vitaly Churkin wrote to all 193 UN members ahead of Wednesday's vote saying that supporting the resolution "will be a serious blow to all attempts to bring the Syrian sides to the negotiating table."
The number of countries that backed the resolution was down from 133 when the last vote on Syria was held in August.
Russia, China, Syria, Iran and North Korea were among 12 countries to oppose the resolution. Fifty-nine countries, including Brazil, South Africa, India and Indonesia abstained.
Syrian troops meanwhile desperately held off rebel offensive on Aleppo's central prison after its wall was blown up by suicide car bombers, the Syrian Observatory for Human Rights said.
"Fierce fighting is taking place within the walls of the compound," said the observatory.
About 4,000 prisoners, including Islamists and common law criminals, are held in the prison on the outskirts of the city, observatory head Rami Abdel Rahman said.
Government forces fired tank shells and carried out air strikes around the jail to repel the rebels, igniting fires and damaging neighboring houses, the observatory said.
Meanwhile, the rebel Free Syrian Army pledged to punish atrocities carried out by its own side, amid outrage over a video showing an alleged insurgent apparently cutting out the heart and liver of a corpse and biting them.
The gruesome footage has raised new fears about the potential for grisly sectarian violence in Syria as the country falls apart.
"Any act contrary to the values for which the Syrian people have paid their blood and lost their homes will not be tolerated," said a statement released by the rebel group.
"The abuser will be punished severely even if they are associated with the Free Syrian Army," it promised.
Field commanders had been instructed "to begin a prompt investigation into the matter in which the perpetrator will be brought to justice."
The man in the video, identified as Khalid al-Hamad, defended his actions in an interview with Time Magazine, saying he had been enraged by footage on the dead soldier's phone showing him "humiliating" a woman and her two daughters.
His actions have been condemned by the Syrian National Coalition, the United States and United Nations.
Syria's internet service collapsed for the second time in less than a week. The SANA state news agency said the blackout was caused by a fiber-optic cable fault.
Internet in the country has regularly been cut, most recently on May 7, with it being restored some 24 hours later.
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