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.
Ban stressed, however, that eliminating chemical weapons from the Syrian conflict "is not a license to kill with conventional weapons."
U.S. Secretary of State John Kerry said the "strong, enforceable, precedent-setting" resolution shows that diplomacy can be so powerful "that it can peacefully defuse the worst weapons of war." Kerry said the destruction of Syria's chemical weapons stockpile will begin in November and be completed by the middle of next year.
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.
As a sign of the broad support for the resolution, all 15 council members signed on as co-sponsors.
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.
Russian Foreign Minister Sergey Lavrov stressed that the resolution does not automatically impose sanctions on Syria.
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 recent flurry of diplomatic activity followed the Aug. 21 poison gas attack that killed hundreds of civilians in a Damascus suburb, and by President Barack Obama's threat of U.S. strikes in retaliation.
After Kerry said Assad could avert U.S. military action by turning over "every single bit of his chemical weapons" to international control within a week, Russia quickly agreed. Kerry and Lavrov signed an agreement in Geneva on Sept. 13 to put Syria's chemical weapons under international control for later destruction, and Assad's government accepted.
Tough negotiations, primarily between Russia and the United States, followed on how Syria's stockpile would be destroyed.
The U.N. resolution's adoption was assured when the five veto-wielding permament members of the Security Council - Russia, China, the United States, France and Britain - signed off on the text on Thursday.
Lavrov told the council that his country will participate in the destruction.
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.
It bans Syria from possessing chemical weapons and condemns "in the strongest terms" the use of chemical weapons in the Aug. 21 attack, and any other use. It also would ban any country from obtaining chemical weapons or the technology or equipment to produce them from Syria.
Kerry stressed that the resolution for the first time makes a determination that "use of chemical weapons anywhere constitutes a threat to international peace and security," which sets a new international norm.
The resolution authorizes the U.N. to send an advance team to assist the OPCW's activities in Syria. It asks Secretary-General Ban to submit recommendations to the Security Council within 10 days of the resolution's adoption on the U.N. role in eliminating Syria's chemical weapons program.
"Syria cannot select or reject the inspectors," Kerry said. "Syria must give those inspectors unfettered access to any and all sites and any and all people."
The resolution requires the council to review compliance with the OPCW's plans within 30 days, and every month after that.
In an indication of the enormity of the task ahead, the OPCW appealed for donations to fund the disarmament, saying it will have to hire new weapons inspectors and chemical experts.
To that end, Britain's foreign minister announced after Friday's vote that the UK would donate $3 million to OPCW Syria Trust fund.
Earlier Friday, Chinese Foreign Minister Wang Yi told the U.N. General Assembly that China was prepared to help fund the disarmament mission.
Associated Press writers Mike Corder in The Hague, Netherlands, Toby Sterling in Amsterdam and Albert Aji in Damascus contributed to this report.