Analysts said the vote showed new resolve to address Syria's conflict, but cautioned it was only a first step and that serious differences between the two sides remained.
The Security Council statement, the first of its kind in two years, urges a political transition and backs a plan to create working groups to discuss ending the war.
It was approved hours after UN officials, including peace envoy for Syria Staffan de Mistura, expressed horror at regime air strikes that killed nearly 100 people in a rebel-held town near Damascus.
The 16-point council statement backed an approach outlined by de Mistura last month after talks with parties to the four-year conflict.
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It proposes four working groups with members from the government and opposition to discuss safety and protection, counterterrorism, political and legal issues and reconstruction.
The council urged "a Syrian-led political process leading to a political transition that meets the legitimate aspirations of the Syrian people".
It called for "an inclusive transitional governing body with full executive powers" to be formed with "mutual consent while ensuring continuity of governmental institutions".
Assad and his backers say he was legitimately elected and cannot be forced aside, but the opposition insists his departure must be part of any peace deal.
The vote follows a flurry of diplomatic activity led by regime backer Moscow, which in recent weeks hosted Syrian opposition figures and officials from Saudi Arabia, a key opposition backer.
The council's vote reflects "an unanimity that is unprecedented for several years," said analyst Karim Emile Bitar, a senior fellow at the IRIS think tank in Paris.
But Bitar said the plan was only a "first step".
"At this stage, neither side is willing to make additional concessions that could allow this rapprochement to have concrete effects on the ground."
Andrey Baklitskiy, an expert at Moscow's PIR-Centre think tank, said there was "nothing revolutionary" in the Security Council statement, and it did not contradict previous Russian statements on the issue.
"The text is fairly evasive, it is not clear on the implementation," he added.