The 15-member council unanimously agreed that the issue of missing Kuwaiti people, property and archives should be dealt with under Chapter 6 of the UN Charter - which urges countries to peacefully resolve any conflicts - instead of Chapter 7.
Chapter 7 of the charter allows the Security Council to authorize actions ranging from sanctions to military intervention if states do not abide by council demands.
The move by the council is a significant political boost for Baghdad.
The Security Council resolution recognized "the importance of Iraq achieving international standing equal to that which it held prior to (1990)." US-led troops drove Iraq out of Kuwait in the 1991 Persian Gulf War.
"This is a new beginning for the relations between our two neighborly and brotherly countries," Iraq's Foreign Minister Hoshyar Zebari told reporters after the vote.
"This is an example for other countries also to resolve their disputes and differences through peaceful means."
The only issues linked to Iraq's invasion of Kuwait that remain under Chapter 7 are an arms embargo and Baghdad's payment of $52 billion (34 billion pounds) in compensation to Kuwait, diplomats say. Iraq still owes $11 billion and has said it expects to pay by 2015.
There are still a range of Chapter 7 issues imposed on Baghdad after Saddam's ouster in 2003, diplomats say, including the freeze and return of Saddam-era assets and trade ban on stolen Iraqi cultural property.
UN Secretary-General Ban Ki-moon has recommended that the UN political mission in Iraq should take responsibility for facilitating the search for missing Kuwaitis, or their remains, property and the country's national archives.