US Secretary of State John Kerry described the resolution as a "strong, enforceable, precedent-setting" document which shows that "diplomacy can be so powerful that it can peacefully defuse the worst weapons of war."
"Provided this resolution is fully implemented, we will have eliminated one of the largest chemical programs on Earth from one of the most volatile places on Earth,” Kerry said.
"Should the regime fail to act, there will be consequences," he added.
The issue of force if Syria is not compliant with the resolution has been a consistent sticking point between the US and Russia.
The resolution "does not allow any automatically enforced punitive measures,” Russian Foreign Minister Sergei Lavrov said after the measure was adopted.
"It confirms agreements reached during the Russian-US meeting in Geneva that any violation of its obligations, as well as the use of chemical weapons by whoever, should become a subject of a detailed consideration by the UN Security Council,” he added.
Lavrov also said the resolution lays basis for a future political and diplomatic solution to the Syrian crisis, which has claimed more than 100,000 lives, according to UN estimates.
He said Russia was ready to take part "in all components” of the future action to eliminate chemical arms in Syria.
Washington has insisted that the threat of military force is crucial to ensuring that the government of Syrian President Bashar Assad abides by the terms of the US-Russia plan to secure and destroy the chemical stockpiles.
Russia has said military intervention is unacceptable.
Under terms of the resolution, the council "decides, in the event of non-compliance with this resolution, including unauthorized transfer of chemical weapons, or any use of chemical weapons by anyone in the Syrian Arab Republic, to impose measures under Chapter VII of the United Nations Charter."
UN Secretary-General Ban Ki-moon hailed the "historic” resolution, calling it "the first hopeful news on Syria in a long time." He also said world powers were aiming to hold an international Geneva-2 peace conference in mid-November.
"As we mark this important step, we must never forget that the catalogue of horrors in Syria continues with bombs and tanks, grenades and guns,” Ban added. "A red light for one form of weapons does not mean a green light for others. This is not a license to kill with conventional weapons. All the violence must end. All the guns must fall silent.”
Syrian UN Ambassador Bashar Ja'afari said Turkey, Saudi Arabia, Qatar, France, the US and other countries that support Syria's rebel forces, should also abide by the resolution.
It was the first time the council formally endorsed a plan for a political transition in Syria under agreements reached during the June 2012 Geneva conference.
Until recently, any Security Council decision-making on Syria was paralyzed when Russia and China blocked US-led efforts to win the UN backing for a military intervention. The two veto-wielding members have blocked three UN Security Council resolutions on Syria since October 2011.
The Security Council vote was put into question earlier in the day Friday, when the executive council of the world's chemical watchdog, the Organization for the Prohibition of Chemical Weapons (OPCW), halted its session to approve a key plan to eliminate Syria's chemical stockpile.
The blueprint was to be incorporated into the resolution.
OPCW spokesman Michael Luhan said one of the delegations to the talks took a pause while waiting for a final response from its capital.
The talks, however, resumed at approximately 2:30 Moscow time [22:30 GMT]. The plan was passed shortly after and entered force immediately.