All member countries of the European Union abstained along with the former Soviet republics of Ukraine, Georgia and Moldova, the statement said.
However, the United States and two small Pacific isles countries voted against the resolution on Tuesday.
The Russian Foreign Ministry stressed that the vote against the resolution was disrespect for those who fought Nazism during the Great Patriotic War of the former Soviet Union.
The resolution, drafted by the Russian delegation, was adopted by the UN General Assembly on Tuesday. The UN General Assembly has annually adopted similar resolutions since 2005.
Russia also said that the authorities of many countries turned a blind eye to the marches of former Nazi veterans and demolitions of the Second World War monuments.
Russia has been especially irritated with the situation in Estonia, Latvia and Lithuania, where former Schutzstaffel veterans regularly assembled for commemorative events. Some state officials in those countries also participated in such events.
However, the U.S. delegation insisted that the Russia-backed resolution contradicts the freedom of gathering and speech.