A draft document agreed by Russia and the US was backed by the 15-member body at a session on New York. It demands the eradication of Syria's chemical weapons but does not threaten automatic punitive action against Syria's government if it does not comply.
The unanimous vote, after two weeks of negotiations, marked a major breakthrough in the paralysis that has gripped the council since the Syrian uprising began. Russia, backed by China, has vetoed three Western-backed resolutions since October 2011 pressuring Assad's regime to end the violence.
A new round of international diplomatic discussions regarding Syria is now also expected to take place in mid-November in Geneva.
Speaking immediately after the vote in New York on Friday, UN secretary-general Ban Ki-moon told the council: "Today's historic resolution is the first hopeful news on Syria in a long time."
But he stressed that eliminating chemical weapons from the Syrian conflict "is not a licence to kill with conventional weapons".
US Secretary of State John Kerry said the "strong, enforceable, precedent-setting" resolution showed that diplomacy can be so powerful "that it can peacefully defuse the worst weapons of war".
Kerry added that the destruction of Syria's chemical weapons stockpile would begin in November.
US President Barack Obama had earlier called the draft UN resolution a "potentially huge victory for the international community".
Foreign Secretary William Hague welcomed the resolution, but also called for greater efforts to end the suffering of its people. He announced that Britain had pledged a £1.8 million contribution to the Organisation for the Prohibition of Chemical Weapons (OPCW), the intergovernmental body which works to ban chemical weapons.
"We expect to have an advance team on the ground [in Syria] next week," OPCW spokesman Michael Luhan said at its headquarters in The Hague, Netherlands.
Following the UN vote, Hague said: "Let us not forget that council action today has come only after two-and-a-half years of unchecked brutality and well over 100,000 dead and millions displaced.
"The failure of the council to tackle the crimes committed on a daily basis has resulted in a culture of impunity in which a brutal regime believed it could get away with murdering its own men, women and children. So it is vital that the council now builds on the consensus we have reached today to make progress today towards a sustainable resolution of the crisis."
The resolution calls for consequences if Syria fails to comply, but those will depend on the council passing another resolution in the event of non-compliance. That will give Russia the means to stop any punishment from being imposed.
The recent flurry of diplomatic activity followed the August 21 poison gas attack that killed hundreds of civilians in Damascus and led to Obama's threatening US strikes against Syria in retaliation.
Former Liberal Democrat leader Sir Menzies Campbell, a member of the House of Commons Foreign Affairs Select Committee, said: "Amidst the euphoria at the UN in New York one significant issue remains unresolved. Is it now accepted, as all the evidence points to, that the Assad regime used chemical weapons against its own citizens?
"Is there a proper determination to bring to justice those who committed a war crime by using these weapons, as the secretary-general of the UN has said? It would be quite wrong and contrary to principle if the result of the rapprochement over Syria between the US and Russia were to wipe a very dirty slate clean."