The Arab-backed initiative, which also calls on President Bashar al-Assad to resign, is the latest of several attempts to bring an end to the crisis.
Syria said the move would only worsen the crisis and encourage "terrorists".
Earlier, China said it was sending a senior envoy to Damascus to negotiate a "peaceful and proper" solution.
The non-binding resolution adopted by the General Assembly backs an Arab League plan aimed at stopping the killings. It was modelled on an earlier resolution in the Security Council that was vetoed by Russia and China.