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 133-12, with 31 abstaining. Syria's ambassador angrily called the vote "a piece of theater."
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 al-Assad resign and a call for other nations to place sanctions on Syria.
Russia and China had objected to those provisions. Both voted "no" Friday, along with Syria, Cuba, Iran, North Korea, Belarus, Bolivia, Nicaragua, Myanmar, Zimbabwe, and Venezuela.
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 United Nations 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 also demands the lockdown of the regime's chemical and biological weapons. Israel's Ambassador Ron Prosor said: "We should not pretend that a regime that cuts the throats of children today will not be prepared to gas them tomorrow. Assad must know that he will be held accountable for using these weapons."
The resolution denounces attacks on children as young as 9 by the Syrian government, military intelligence services, and militias. It 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.
Syrian Ambassador Bashar Ja'afari called the resolution's main sponsors, Saudi Arabia, Qatar, and Bahrain, "despotic oligarchies."