The vote after two weeks of intense negotiations marked a major breakthrough in the paralysis that has gripped the council since the Syrian uprising began. Russia and China previously vetoed three Western-backed resolutions pressuring President Bashar Assad's regime to end the violence.
"Today's historic resolution is the first hopeful news on Syria in a long time," U.N. Secretary-General Ban Ki-moon told the council immediately after the vote.
For the first time, the council endorsed the roadmap for a political transition in Syria adopted by key nations in June 2012 and called for an international conference to be convened "as soon as possible" to implement it.
Ban said the target date for a new peace conference in Geneva is mid-November.
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 Assad ally Russia the means to stop any punishment from being imposed.
The vote came just hours after the world's chemical weapons watchdog adopted a U.S.-Russian plan that lays out benchmarks and timelines for cataloguing, quarantining and ultimately destroying Syria's chemical weapons, their precursors and delivery systems.
The Security Council resolution enshrines the plan approved by Organization for the Prohibition of Chemical Weapons, making it legally binding.
The agreement allows the start of a mission to rid Syria's regime of its estimated 1,000-ton chemical arsenal by mid-2014, significantly accelerating a destruction timetable that often takes years to complete.
"We expect to have an advance team on the ground (in Syria) next week," OPCW spokesman Michael Luhan told reporters at the organization's headquarters in The Hague, Netherlands immediately after its 41-member executive council approved the plan.
The U.N. resolution's adoption was assured when the five veto-wielding permanent members of the Security Council - Russia, China, the United States, France and Britain - signed off on the text on Thursday.
As a sign of the broad support for the resolution, all 15 members signed on as co-sponsors Friday.
Russia and the United States had been at odds over the enforcement issue. Russia opposed any reference to Chapter 7 of the U.N. Charter, which allows for military and nonmilitary actions to promote peace and security.
The final resolution states that the Security Council will impose measures under Chapter 7 if Syria fails to comply, but this would require adoption of a second resolution.