The document will be signed later by the Russian president.
The chairman of the culture committee, Alexander Dzasokhov, said: "The stained glass windows are unique, and have importance for history, art and science."
The senator added that Germany will cover the costs for transportation, security and authentication of the glass panels.
In 2002, Russia returned 111 stained glass panels from the 20-meter-high altar to St. Mary's Church, Marienkirche, in Frankfurt-upon-Oder, which had been stored in the Hermitage, the country's main museum in St. Petersburg.
Authorities later discovered that Moscow's Pushkin Museum of Fine Arts had a further six panels in their stores.
Russia and Germany have pledged to return valuable artworks captured during World War II. The restitution process, however, has proceeded slowly, as many in Russia consider the artifacts compensation for the country's wartime losses, and both sides are anxious about parity in returning looted cultural objects.
Nazis looted artworks throughout Europe and the occupied parts of the then Soviet Union, secreting many of them into Nazi leaders' private collections.
After the war, Soviet security bodies moved to return as many of them as possible also confiscating other cultural objects as military trophies.
The two countries have since made up impressive lists of displaced valuables as part of their restitution pledges. The actual volume of cultural objects looted by Germany and Russia is unknown as many of them were destroyed or stolen by individual soldiers.
One of the most famous objects lost in the war was the legendary Amber Room, made in Prussia in the early 18th century, and believed to have been looted by Nazi troops in 1945.
In February, media reported Russia's lost treasure may have been discovered by treasure hunters at an underground site near the German border with the Czech Republic. The excavation recently ground to a halt, however, after a disagreement between the people leading the dig.