Russia, China and Iran were among the 12 countries that opposed the resolution on Wednesday.
Russia called the resolution, co-sponsored by the United States, "counterproductive and irresponsible."
The resolution was adopted by a vote of 107-12 with 59 abstentions. Argentina, Brazil, and more than a dozen other Latin American and Caribbean countries abstained from voting.
Russian Deputy Ambassador to the UN Alexander Pankin called the resolution "very harmful and destructive," saying it disregards "illegal actions of the armed opposition." He also accused the resolution’s Arab sponsors of attempting to replace the Syrian government instead of trying to find a political solution to the crisis in Syria.
Syrian Ambassador to the UN Bashar Ja'afari also stated that the resolution "seeks to escalate the crisis and fuel violence in Syria.”
The non-binding resolution, which was drafted by a number of Arab states, calls for a “political transition” and refers to the foreign-backed militants in Syria as "effective representative interlocutors” needed for the transition.
The Syria crisis began in March 2011, and many people, including large numbers of soldiers and security personnel, have been killed in the violence.
The Syrian government says that the chaos is being orchestrated from outside the country, and there are reports that a very large number of the militants are foreign nationals.
Damascus says the West and its regional allies, such as Qatar, Saudi Arabia, and Turkey, are supporting the militants.
In an interview recently broadcast on Turkish television, Syrian President Bashar al-Assad said that if the militants take power in Syria, they could destabilize the entire Middle East region for decades.
“If the unrest in Syria leads to the partitioning of the country, or if the terrorist forces take control… the situation will inevitably spill over into neighboring countries and create a domino effect throughout the Middle East and beyond,” he stated.