The 193-nation UN General Assembly passed the non-binding resolution with 100 votes in favor, 11 against, and 58 abstentions.
Armenia, Belarus, Bolivia, Cuba, North Korea, Nicaragua, Sudan, Syria, Venezuela and Zimbabwe joined Russia in rejecting the document.
"The result is rather satisfying for us as we have won a moral and a political victory,” Vitaly Churkin said.
"It clearly shows that Russia is not isolated [by the international community],” the diplomat said.
In the wake of the Ukrainian crisis, Crimea sought reunification with Russia, a move backed by over 96 percent of voters in the March 16 referendum.
The region, previously an autonomous republic within Ukraine, became part of Russia last week.