The windows come from the Marienkirche in Frankfurt an der Oder near the Polish border.
In 2002, 111 stained glass windows from the same church were returned to Germany but the six others were discovered later in Moscow's Pushkin Museum, home to numerous art items removed by Soviet forces at the close of the war.
Germany is to pay the cost of the removal, the head of parliament's culture committee, Grigory Ivliyev, told the news agency. He added that Germany is engaged in restoring a number of cultural treasures in Russia damaged during the war, including a church in the western Novgorod region.
Russian law allows for the return of cultural treasures if they were not used for "militarism and Nazism," RIA Novosti said.