An ABC News/BBC/NHK poll in August found 62 percent want to keep one unified Iraq with a central government in Baghdad.
The federal approach suggested in the Biden-Brownback Senate resolution gets only 28 percent support from Iraqis. Only Kurds would prefer to not keep the current structure.
Warner called the bill's passage an "extraordinary moment, because it marks the high-water mark of all the many debates and resolutions we've had in terms of bipartisanship."
Twenty-six Republicans voted for the measure, along with 47 Democrats and two independents. Two other presidential candidates, Sens. John McCain, R-Ariz., and Barack Obama, D-Ill., did not vote.
In May 2006, Biden and Leslie Gelb, president emeritus of the Council on Foreign Relations, proposed the federalist solution in a New York Times op-ed.
"The idea, as in Bosnia, is to maintain a united Iraq by decentralizing it, giving each ethno-religious group -- Kurd, Sunni Arab and Shiite Arab -- room to run its own affairs, while leaving the central government in charge of common interests," they wrote. "We could drive this in place with irresistible sweeteners for the Sunnis to join in, a plan designed by the military for withdrawing and redeploying American forces, and a regional nonaggression pact."
Before Wednesday's vote, Biden aides anticipated fewer than 30 senators would support his proposal. The senator's longshot second run for the White House is built largely on the notion that he is the only candidate thinking seriously about solutions to issues surrounding the Iraq War.
ABC News' Jennifer Duck and Peyton Craighill contributed to this report.