It now goes to President Vladimir Putin to sign or turn down.
The bill is one part of a larger measure by lawmakers retaliating against a recently signed U.S. law that calls for sanctions against Russians deemed to be human rights violators.
Some top government officials, including the foreign minister, have spoken against the bill, arguing that it would be in violation of Russia's constitution and international obligations.
Several people protesting the bill were detained outside the Federation Council on Wednesday morning.
Critics of the bill say it victimizes orphans by depriving them of an opportunity to escape often-dismal Russian orphanages. There are about 740,000 children without parental custody in Russia.