The treaty makes it a crime for would-be terrorists to possess or threaten to use nuclear weapons or radioactive material.
A resolution on Wednesday adopted by the 191-member world body by consensus calls on all countries to sign and ratify the International Convention for the Suppression of Acts of Nuclear Terrorism.
The treaty will be opened for signatures on 14 September and must be ratified by 22 countries to come into force.
"By its action today, the General Assembly has shown that it can, when it has the political will, play an important role in the global fight again terrorism," US deputy ambassador Stuart Holliday told delegates after the vote.
"The nuclear terrorism convention, when it enters into force, will strengthen the international legal framework to combat terrorism."
Russia under the presidency of
Boris Yeltsin proposed the treaty
Russia's deputy UN ambassador Alexander Konuzin, whose country sponsored the resolution, hailed its approval, saying "it's the first time that an anti-terrorist convention has been developed on the basis of preventing - that is not after the fact but before the terrorist acts which are criminalised by this convention".
The treaty makes it a crime for any person to possess radioactive material or a radioactive device with the intent to cause death or injury, or damage property or the environment.
It would also be a crime to use such material or devices to damage a nuclear facility.
A person would also commit a criminal act by threatening to use radioactive material or devices - or unlawfully demanding nuclear material or other radioactive substances.
Accomplices and organisers would also be covered by the convention.
"By its action today, the General Assembly has shown that it can, when it has the political will, play an important role in the global fight again terrorism"
US Deputy Ambassador
Countries that are parties to the treaty would be required to make these acts criminal offences under their national laws, "punishable by appropriate penalties which take into account the grave nature of these offences".
Russia launched the campaign for a treaty to combat nuclear terrorism more than seven years ago, when Boris Yeltsin was president, but it was stymied for years because countries believed the draft convention was trying to define terrorism - an issue that has deeply divided the UN.
Diplomats said the roadblock was broken after the drafting committee's last formal meeting in November, when the Organisation of Islamic Conference decided that the new treaty could focus on criminalising specific actions related to nuclear terrorism as other anti-terrorism treaties have done.