The non-binding resolution endorsed the Arab League's effort for an end to the Assad government's deadly crackdown on the nearly yearlong pro-democracy uprising and asking the president to leave office.
Those voting against the measure included Russia and China, CNN reported. There were 17 abstentions.
Russia and China had vetoed a similar effort this month in the U.N. Security Council, which would have made it binding. Besides Russia and China, other permanent members of the Security Council are the United States, France and Britain.
The General Assembly vote was thus seen as showing where the rest of the world stands on the issue.
"We hope that the regime will listen to this overwhelming message from the international community today," British Ambassador Mark Lyall Grant said in a statement.
"Today, the U.N. General Assembly sent a clear message to the people of Syria: the world is with you," Susan Rice, U.S. ambassador to the world body said in a statement.
Rice said Assad "has never been more isolated. A rapid transition to democracy in Syria has garnered the resounding support of the international community. Change must now come."
French Foreign Minister Alain Juppe said, "For France, this is a new step towards the end of the martyrdom of the Syrian people."
Reacting strongly against the resolution, Syria's U.N. Ambassador, Bashar Jaafari, called the League of Arab States "broken, both politically and morally."
"If things continue in this manner ... the United Nations will collapse -- morally first and entirely second," Jaafari said.
Asked if there would be a temporary cease-fire in the violence-hit town of Homs, the Syrian ambassador was quoted as saying: "Ceasefire? We are not in a civil war! We are not in an armed conflict!"
The Assad government has denied any crackdown, saying armed gangs and foreign fighters are seeking to destabilize the government. Thousands have died in the uprising.
Though non-binding, the General Assembly resolution is significant humiliation for Assad, The New York Times said.
Iran, Venezuela and North Korea opposed the resolution, calling it an unwarranted interference in Syria's internal politics.
The Times report said the resolution had more than 70 co-sponsors.
The resolution called for Assad to transfer power to a vice president and for negotiations among the antagonists for forming a new government.
On Thursday, the Times quoted Syrian opposition groups as saying the military campaign had expanded, with government forces hitting targets in Idlib Province in the north and Hama Province in central Syria, as well as military attacks farther south.