Thirteen of the 15 countries on the council voted in favor of the resolution, the United Nations News Center said. But Russia, one of the five permanent veto powers on the council, opposed it, and China, another permanent member, abstained.
Crimeans are scheduled to vote Sunday. Critics, including the United States and European Union, say the hastily scheduled referendum will not be valid.
Vitaly Churkin, the Russian representative on the council, said his country believes the decision should be in the hands of Crimeans. He said the goal of the resolution was "to declare illegal the planned March 16 referendum where residents of the Republic of Crimea should decide on their future."
The resolution called on other countries to respect the "sovereignty, independence, unity and territorial integrity" of Ukraine.
In Moscow, several thousand people gathered for a rally opposing Russia's intervention in Crimea. Another demonstration supported Russian action.
Media reports said 30,000 people attended the rally opposing the government, while police said there were only 3,000 in the crowd, RIA Novosti reported. They carried Russian and Ukrainian flags and chanted "Ukraine, we are with you."
Police said several people spotted with Nazi flags were arrested.