The UN condemned the suicide attacks the next day
"Above all, it provides a legal foundation for forming a coalition against terrorism," said Russian Foreign Ministry Alexander Yakovenko on ORT state television.
The BBC Moscow correspondent says Russia's approval is a signal that it will support moves which go through the UN, but would be less happy with any unilateral action from the United States.
High level security meetings have been held in Moscow with American officials on Washington's global campaign against terrorism.
The resolution
The United States-sponsored resolution also calls on all states to co-operate on exchanging information about the activity of terrorist groups.
President Bush's administration has made it clear that it sees the closing down of funding for terrorism as one of key planks in its campaign after the 11 September attacks on New York and Washington in which more than 6,000 people were killed.
Those who do not abide by the resolution might find themselves subject to UN sanctions.
One stumbling block was the definition of a "terrorist," which is not contained in the resolution and could be added later by the 189-nation General Assembly.
"There is a huge grey area of what is a terrorist," said one council diplomat.
Closing loopholes
United Nations diplomats say the resolution passed late on Friday should be seen as a practical way of closing loopholes that exist in current UN anti-terrorism conventions, many of which have not been signed or ratified by member states.
The resolution invokes the enforcement rules of Chapter Seven, which obliges all states to implement it immediately.
There is also a provision for the establishment of a Security Council committee that would monitor whether countries are actually abiding by the rules of the resolution, drawing on the expertise of banking and other technical experts.
It also calls on states to exchange information to prevent the movement of terrorists across international borders.
The new resolution follows a call this week by President Bush for nations to freeze the assets of people and groups linked to terrorism.
But the resolution does not mention the list of 27 individuals and groups whose assets were frozen by the US.
The council's permanent members - France, China, Russia, the United States and Britain - agreed on the general terms of the resolution before it was introduced to the rest of the council, diplomats said.