The name of the mission was omitted from the text of the resolution in connection with demands from Abkhazia, a former Georgian republic recognized as an independent state by Russia.
Abkhazia had threatened not to allow UN monitors to continue working unless the mission's name reflected its independent status.
The current mandate was set to expire on Sunday.
The UN mission in the Georgia-Abkhazia conflict zone - formerly known as the United Nations Observer Mission in Georgia (UNOMIG) - was deployed in 1993 to oversee a ceasefire following a bloody post-Soviet conflict that resulted in Abkhazia gaining de facto independence.
Moscow recognized Abkhazia and South Ossetia after a five-day war last August with Georgia, which attacked the latter in an attempt to bring it back under the control of Tbilisi. Residents of both Abkhazia and South Ossetia have held Russian citizenship for many years.
Russia's move was condemned by the United States and Europe. Nicaragua has so far been the only other country to follow Russia in recognizing South Ossetia and Abkhazia.
Russia's UN ambassador said on Friday the new resolution contained no mention of Georgia's territorial integrity.
"The text contains no mention of Georgia's territorial integrity, which speaks volumes," Vitaly Churkin said.