Iraq said it will consider whether to adopt the resolution
The vote came at the last moment before the current phase is due to expire at midnight on Friday.
The United States in turn agreed to review a December 1999 resolution that would ease sanctions in return for increased cooperation with UN weapons inspectors.
US deputy ambassador James Cunningham told the Associated Press news agency that the US was satisfied with the resolution.
"I think that, starting on 1 June, there will be a new system in place that will enable us finally to achieve... a greatly improved oil-for-food programme and greatly facilitated implementation of the sanctions regime," he said.
Iraq's UN ambassador, Muhammad al-Douri, said that the country will consider whether to accept the deal once the UN formally adopts it.
Weapons inspection
The 11-year-old sanctions against Iraq, put in place after the Gulf War, can only be removed after UN inspectors can determine for certain that Iraq's weapons of mass destruction have been dismantled.
The UN inspectors left Iraq's capital Baghdad in December 1998 shortly before US-British air strikes in the country, and Iraq has forbidden them to return until the sanctions are lifted.
The United States and Russia have often disagreed over policy towards Iraq.
Russia has close ties with Iraq, including several lucrative oil contracts, while America has consistently demanded that the country halt its weapons programmes.
It has also warned that Iraq may face unspecified consequences should UN inspectors not be permitted to enter the country.
Differences suspended
However, the war in Afghanistan has altered such disagreements, as Russia has become an integral part of the coalition against terror.
British ambassador Sir Jeremy Greenstock told Reuters news agency the deal enables differences over the sanctions to be suspended at least until next year.
"Because of what's going on in Afghanistan we are taking this in stages," he said.
"While everything else is going on there is no need to have a fight on this."