The Security Council vote came shortly after the executive committee of the Organization for the Prohibition of Chemical Weapons voted to fast-track Syria into the Chemical Weapons Convention, CNN reported.
"This is a breakthrough arrived at through hard-fought diplomacy," a senior State Department official told CNN.
"The Russians have agreed to support a strong, binding and enforceable resolution that unites the pressure and focus of the international community on the Syrian regime to ensure the elimination of Syria's chemical weapons," the official said.
"We'll have to be vigilant about following through, but this could be a significant victory for the international community, and demonstrate how strong diplomacy can allow us to secure our country and pursue a better world," U.S. President Barack Obama said shortly before the Security Council vote.
The OPCW will send a team of inspectors to Syria Monday, a spokesman for the group told CNN.
Obama noted in earlier remarks the United States and Russia arranged an agreement, partnering with the P-5 (Germany, the United States, Russia, China, France and the U.N. Security Council), to see Syria's chemical weapons destroyed by the international community.
"This resolution will require the destruction of a category of weapons that the Syrian government has used ruthlessly and repeatedly against its own people. And this resolution will make clear that there are going to be consequences for non-compliance," U.S. Ambassador to the United Nations Samantha Power said earlier, referring to the U.N. resolution.
Power called the move significant because it is the first time since the start of Syria's civil war more than two years ago that the U.N. Security Council has imposed binding obligations on President Bashar Assad's regime.
The U.N. mission investigating the alleged use of chemical weapons in Syria continued working on a report it hopes to deliver by late October, it said in a statement Friday from Damascus. It said it intends to finalize its activities in Syria by Sept. 30.
Russian U.N. Ambassador Vitaly Churkin had said Thursday he hoped the resolution would be adopted soon, "maybe even tomorrow night."
Russia, a main ally of Assad, blocked earlier anti-Syria resolutions presented to the U.N. Security Council.
A draft obtained by CNN indicated the resolution asks the OPCW director general and the U.N. secretary-general report non-compliance to the Security Council. If there is non-compliance, the council would impose measures under a Chapter VII of the U.N. Charter.
The resolution does not authorize the automatic use of force if Syria is said to be in violation, language that had been sought by the United States, CNN said.
The United States and other Western nations blame the Syrian government for an Aug. 21 chemical weapons attack in Damascus' suburbs that U.S. officials estimate killed 1,400 people. Syria and Russia have countered that they think rebels used the weapons.