Some delegations, including Syria, Cuba and Venezuela, however criticised the text's failure to include a clear legal definition of terrorism and to specifically condemn state terrorism.
The text resolves to "consistently, unequivocally and strongly condemn terrorism in all its forms and manifestations, committed by whomever, wherever and for whatever purposes, as it constitutes one of the most serious threats to international peace and security."
The resolution, adopted only a few days before the 5th anniversary of the September 11 attacks on the United States, was to serve as a basis for the conclusion of a "comprehensive" UN convention to curb international terrorism.
For nine years, diplomats have been sweating over a draft convention that would encompass previous texts on the fight against terrorism, including bombings, financing and nuclear and biological threats.
But they have so far stumbled over an acceptable definition of terrorism, particularly in the Middle East, where groups seen by some countries as terrorists are viewed by others as freedom fighters.
The resolution makes clear that "any measures that we undertake to prevent and combat terrorism must comply with our obligations under international law, including the Charter of the United Nations and relevant international conventions and protocols, in particular human rights law, refugee law and international humanitarian law."
But Syria's UN ambassador Bashar al-Jaafari said although he backed the text in the interest of the general consensus, he deplored the fact that it did not specifically mention state terrorism, citing as an example Israeli attacks against Lebanon and the Palestinians.
He also said the text failed to make a distinction between terrorism and the legitimate right of people under occupation to fight for their freedom and also failed to address the root causes of terrorism.
He called the resolution "unbalanced with many faults and shortcomings."
"A legal definition of terrorism is a precondition to implementing this strategy," Mr Jaafari said.
Cuba also warned that combatting terrorism should not be a "pretext to interfere in the internal affairs of other states."
UN chief Kofi Annan, who was present, said adoption of the strategy "sends a clear message that terrorism is unacceptable, no matter who commits it, no matter what the reason."
"The test will be how they (assembly members) implement it, he told reporters.
The UN counter-terrorism strategy is articulated around what Mr Annan called the "five Ds":
- Dissuading people resorting to, or supporting, terrorism
- Denying terrorists the means to carry out an attack
- Deterring states from supporting terrorism
- Developing state capacity to defeat terrorism
- Defending human rights