The non-binding resolution also condemns “continued and widespread violations of human rights” in the country.
The text offers full support for the Arab League’s plan for peaceful political transition in Syria, a plan that calls for President Assad to hand over power to his deputy.
Western diplomats hoped the high number of “yes” votes would send a strong signal to President Assad’s regime about the strength of opinion in the international community.
Foreign Secretary William Hague said: “The message is unambiguous. The violence must stop immediately.”
He added: "I look forward to discussing how we can support the Arab League further at the Friends of Syria meeting next week."
Syria’s ambassador to the UN said the resolution would contribute further chaos and crisis in his country, and accused the UN of meddling in its internal affairs.
He maintained the government was taking steps to address the legitimate concerns of the Syrian people, and did not need any intervention from outside.
As well as Syria, 11 countries including Russia, China, and Iran, voted against the resolution. North Korea, Belarus, Zimbabwe and Cuba also opposed it. There were 17 abstentions.
Russia had attempted to water the resolution down by introducing amendments but they were rejected by Arab sponsors of the original draft.
After the vote, US ambassador to the UN Susan Rice tweeted: "The UN General Assembly has just sent a clear message to the people of Syria: The world stands with you."
Top Russian diplomat, Gennady Gatilov, said the draft was too close to the UN Security Council resolution it vetoed along with China two weeks ago, which backed an Arab League call for President Bashar al Assad to quit.
"We can't vote for that resolution, because it still remains unbalanced," Gatilov said. "It directs all the demands at the government, and says nothing about the opposition."
France indicated the Arab-led initiative at the Assembly was proposed in order to establish humanitarian channels into the country to ease the suffering of the population, and not – contrary to Russia's argument – to enact regime change.
Opening the discussions, Syria's envoy to the United Nations Bashar Ja'afari repeated his government's position regarding "terrorist attacks in Syria," calling on other members to "stop adding fuel to the fire of violence and sectarian strife in Syria".
Ahead of the vote, UN chief Ban Ki-moon renewed his calls for an end to the violence in Syria and urged the international community to find a common response to the unrest.
"What is important at this time is that first the Syrian authorities must stop killing their own people," he said.
"This violence should stop from all sides whether by national security forces or by opposition forces.
"I urge the international community to speak in one voice: stop the violence. Stop the bloodshed. The longer we debate, the more people will die."
New video of the onslaught in Homs shows buildings collapsing and large parts of the city in ruins.
Activists say residents in the country's most rebellious city are going without food and electricity, and are struggling to cope after nearly two weeks of intense bombardment and shelling.
International rights watchdogs have said over 400 people have been killed in Homs since February 4. Tanks continue to roll down streets lined with wrecked homes and businesses.
Another video has emerged of an explosion in the Syrio-Lebanese border town of Zabadani which has also come under fierce assault by government forces fearful of Free Syrian Army personnel operating so close to the capital Damascus.
One activist in Deraa – the town in which protests against the president began almost a year ago – described the attacks as "very methodical," saying regime forces were attacking the province "village by village".
"The (rebel) Free Syrian Army is trying to push them back but it is not equipped and is forced to retreat. Regime troops are taking revenge on residents," he said.
As the country-wide crackdown goes on, prominent blogger and symbol of the revolution Razan Ghazzawi and other top activists have been arrested.
Ghazzawi, as well as prominent human rights activist Mazen Darwish, his wife and 11 others were taken by security forces during a raid on the Syrian Centre for Media and Freedom of Expression, which is located in central Damascus and is headed by Darwish.
Known as @RedRazan on Twitter, her knowledge of the rebel network, access to its organisers and contact with the West makes her an important voice for many in the opposition.
It is the second time the American-born activist has been detained. In December her arrest on the Jordanian border sparked a massive outcry from Syria's international activist network.
"We at the Syrian Centre for Legal Studies condemn these arrests and call on Syrian authorities to immediately release them," human rights lawyer Anwar Bunni said.
Amid the violence, Russia has sought to water down the UN's non-binding resolution and welcomed the regime's pledge to hold a referendum on a new constitution.
According to diplomats, Moscow has requested that a paragraph referring to the Arab League plan, which would have seen Syria's vice president replace Mr Assad, be changed.
Another amendment would link the return of Syrian troops to their barracks to an "end of attacks by armed groups against state institutions".
Russia also wants the opposition "to dissociate themselves from armed groups engaged in acts of violence", and not mention Syrian government abuses against civilians.
The Arab countries responsible for bringing the resolution to the General Assembly reportedly rejected the Russian requests, but Moscow could still seek to have them inserted when the issue is debated.
Russian foreign minister Sergei Lavrov has ruled out any UN resolution "that could legitimise regime change".
He also said Mr Assad's offer to hold a referendum in two weeks on the constitution, which would lead to elections in 90 days, was "a step forward".
"It is coming late, unfortunately, but better late than never," Mr Lavrov commented.
But others poured scorn on the proposed constitution and elections, which would establish a multi-party system and limit the president to two terms of seven years.
Melhem al Droubi, a member of the opposition Syrian National Council and the Muslim
Brotherhood, said: "The truth is that Bashar al Assad has increased the killing and slaughter in Syria.
"He has lost his legitimacy and we aren't interested in his rotten constitutions, old or new."
The Local Coordinators Union (LCC) also issued a statement, saying: "The draft constitution is no more than a political tool or a policy paper written by the barbaric regime… We see no alternative but to topple the regime."
The United States called the move "laughable". White House spokesman Jay Carney said: "Promises of reforms have been usually followed by increase in brutality and have never been delivered upon by this regime since the beginning of peaceful demonstrations in Syria."
The Syrian military continued its offensive against the centres of opposition even as the government announced the referendum.
A fresh attack was launched on Hama with anti-aircraft guns mounted on the back of armoured vehicles firing into residential neighbourhoods of the city, opposition activists said.Post published in: News