Before the vote, Secretary-General Ban Ki-Moon reminded the Assembly of the fresh violence in the city of Aleppo and drew comparisons between the failure to act in Syria with the international community's failure to protect people from past genocide in Srebrenica and Rwanda.
"The conflict in Syria is a test of everything this organization stands for," Ban said. "I do not want today's United Nations to fail that test."
The vote came after the more powerful Security Council was stopped by a series of Russian and Chinese vetoes on resolutions that would have opened the door to sanctions on Syria.
The General Assembly vote was 133 in support of the resolution and 12 against, with 31 abstaining. Though General Assembly resolutions are unenforceable, a strong vote can carry moral weight.
Even so, the resolution's Arab sponsors this week weakened two key provisions -- a demand that President Bashar Assad resign and a call for other nations to place sanctions on Syria.
Russia and China had objected to those provisions.
The revised resolution takes a swipe at Russia and China by "deploring the Security Council failure" to act.
Frustration over the lack of action was clear. Former U.N. chief Kofi Annan resigned Thursday as the joint U.N.-Arab League envoy to Syria after his peace proposals failed. Friday's session rang with accusations over why Annan's mission failed.
The Syria uprising has left 19,000 dead since it erupted in March 2011. The U.N. estimates that 1.5 million people have been forced to abandon their homes but remain in the country.
"The acts of brutality that are being reported may constitute crimes against humanity or war crimes," Ban said of the Aleppo fighting. "Such acts must be investigated and the perpetrators held to account."
The resolution backs Annan's "demand that the first step in the cessation of violence has to be made by the Syrian authorities."
It denounces attacks on children as young as 9 by the Syrian government, military intelligence services and militias, as well as "killing and maiming, arbitrary arrest, detention, torture and ill-treatment, including sexual violence, and use as human shields."
It also condemns the increasing Syrian military reliance on heavy weapons, including tanks and helicopters, and "failure to withdraw its troops and heavy weapons to their barracks" in line with Annan's proposals.
It also demands the lockdown of the regime's chemical and biological weapons.
Syrian Ambassador Bashar Ja'afari reacted angrily to the resolution, calling its main sponsors, Saudi Arabia, Qatar and Bahrain, "despotic oligarchies."
Another likely victim of the Security Council stalemate are the U.N. military observers who have been monitoring the spiraling violence in Syria.
The mission is in the midst of a 30-day extension of its mandate, which expires on Aug. 19. Extending it would require passage of another resolution in the Security Council.
The mission has been largely kept from its work by the violence, and it is already being cut back, from its original authorized strength of 300 to currently 115 monitors and 80 civilians.
