Tuesday's resolution, introduced by Republican Senator Lindsey Graham and backed by senators of both parties, encouraged the International Criminal Court (ICC) in The Hague and other nations to target the Russian military in any investigation of war crimes committed during Russia's invasion of Ukraine.
"All of us in this chamber joined together, with Democrats and Republicans, to say that Vladimir Putin cannot escape accountability for the atrocities committed against the Ukrainian people," Democratic Senate Majority Leader Chuck Schumer said in a speech on the Senate floor ahead of the vote.
Russia calls its actions a "special military operation" to demilitarise and "denazify" Ukraine.
Putin has also called the country a US colony with a puppet regime and no tradition of independent statehood.
Moscow has not captured any of the 10 biggest cities in the country following its incursion that began on February 24, the largest assault on a European state since 1945.
Earlier on Tuesday, Russia had put President Joe Biden, Secretary of State Antony Blinken and other top officials on a "stop list" that bars them from entering the country.
Their names, together with Defence Secretary Lloyd Austin, CIA chief William Burns, National Security Adviser Jake Sullivan and others, were included on a list of 13 individuals banned from Russia in response to sanctions imposed by the US on Russian officials.
Former Secretary of State Hillary Clinton was also named, as was Biden's son Hunter.
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