The resolution was passed with 77 delegations voting in favor and six against, with 74 abstentions.
Court hearings on the matter could take more than a year, delaying Kosovo's integration into international society.
Meanwhile, Kosovo's Prime Minister Hasim Taci said the sovereignty of Kosovo had been decided and there was nothing Belgrade could do to influence or change it.
"Independence is a closed matter and no initiative from Belgrade could talk us out of it except for some kind of supernatural force," Taci said.
Speaking at a UN General Assembly session on September 23, Serbian President Boris Tadic said: "To vote against such a resolution would mean to reject the right of any country - now and in the future - to seek legal solutions within the framework of the UN. Such an approach could lead to the end of the UN system as we know it today."
The status of Kosovo has been a point of international disagreement since NATO's bombings of the former Yugoslavia in 1999 ended a bloody conflict between Kosovo Albanians and Serb forces. Russia, Serbia's long-time ally, has been an outspoken supporter of Belgrade's position on Kosovo.
Kosovo declared independence in February and has been recognized by 48 UN member states out of 192, while Serbia and Russia insist that it will always remain a part of Serbia.
Kosovo authorities have no control over about 15 percent of the region, populated by some 200,000 Serbs, who have vehemently opposed UN or EU administration rule.
