The vote in the Senate was 61-33.
House Speaker John Boehner of Ohio has not decided whether to consider the bill before the current Congress ends on Jan. 3, Brendan Buck, a spokesman for Boehner, said in an email.
New York Gov. Andrew Cuomo urged the House to take up the Sandy aid package as part of its agenda.
He said action on the package should not be affected "by the same partisan contention or parliamentary process" holding up a deal on avoiding the "fiscal cliff" of tax increases and spending cuts set to take effect in January.
"Our demonstrated need and the House's past precedent should make this vote a slam dunk," Cuomo said.
New Jersey lawmakers said the bill would go a long way toward helping their battered state recover.
"This bill is a winner for New Jersey," Democratic Sen. Frank Lautenberg said.
He and Democratic Sen. Robert Menendez urged swift House action, saying their work is not yet done.
"This is not a partisan matter, nor is it limited to just the Northeast," Menendez said. "Passage of Sandy relief is of great national concern."
The Senate approved minor changes to the bill by voice vote on Friday.
One would require the Department of Homeland Security redeploy vehicles rather than purchase new ones to replace those damaged by salt water from the storm surge. Another change would authorize federal money to be spent on increasing security at overseas embassies.
The Senate also agreed to require that Congress and the public be notified about certain grant recipients and to bar Hurricane Sandy aid money from going to "tax cheats, deceased individuals" or people with "serious delinquent tax debts." Both amendments were authored by Republican Sen. Tom Coburn of Oklahoma.
An amendment to reduce the $336 million in funding for Amtrak was withdrawn because it could not pass, according to Republican Sen. John McCain of Arizona, who sponsored the proposal.
"Amtrak's too strong a lobby," McCain said in an interview.
The Senate also rejected proposals that would have substantially reduced the scope of the emergency aid for Sandy victims, including a Republican alternative that would have reduced the overall amount to $23.8 billion by concentrating on immediate costs through the end of March.
The Senate-passed bill includes:
-- $9.7 billion in new borrowing authority for the National Flood Insurance Program to pay claims filed by homeowners and businesses.
-- $11.5 billion for the Federal Emergency Management Agency's Disaster Relief Fund.
-- $10.8 billion for the Transportation Department's Public Transportation Emergency Relief fund for assistance to subway, commuter rail and public bus systems.
-- $17 billion for Community Development Block Grants administered by the Department of Housing and Urban Development, with $15 billion designated for disaster relief and restoration and the other $2 billion for mitigation to reduce future risks.
-- $5.35 billion for Army Corps of Engineers infrastructure projects including $3.5 billion for mitigation projects to reduce future flooding risks.
-- $760 million for the Small Business Administration's disaster loan program.
-- $921 million for highway and bridge repairs funded through the Federal Highway Administration.
(Copyright © 2012 USA TODAY)