The Senate quickly adopted the measure on a unanimous voice vote after the House approved the bill on a 354-67 vote.
The House action came after Speaker John Boehner was raked for not holding a vote on a $60 billion relief package earlier this week.
The bill approved Friday allows the Federal Emergency Management Agency to pay out claims to those who have federal flood insurance, The Washington Post reported.
FEMA had said the National Flood Insurance Program likely would run out of money next week. Nearly 140,000 Sandy-related flood insurance claims have been filed, FEMA said, with most still not paid or not paid in full.
Boehner did not bring a Senate-approved $60 billion relief measure to the floor Tuesday, drawing sharp criticism from New Jersey and New York leaders from both parties. Among other things, the House leadership cited discontent in its caucus related to the "fiscal cliff" deal that, among other things, raised tax rates for the wealthiest Americans.
All the votes against the $9.7 billion bill Friday were Republican, the Post said.
Among those voting no was Rep. Paul Ryan, the Budget Committee chairman and 2012 Republican vice presidential nominee, The Hill reported.
"This legislation proposes to increase the program's borrowing authority by $9.7 billion," Ryan said during debate. "It would be irresponsible to raise an insolvent program's debt ceiling without making the necessary reforms."
The dissenters were encouraged by the conservative Club for Growth, which had argued the disaster spending should be offset with cuts to other government programs.
Observers said continued GOP opposition could pose problems for a larger $51 billion Sandy relief package Boehner has promised will come before the House Jan. 15.
Republicans from affected states had threatened to withhold support for Boehner as speaker of the just-convened 113th Congress. Boehner won with 220 votes, despite a threat of a coup and several defections.
Still, the Post said, Democrats criticized Boehner for the delay, noting the $60 billion bill passed on a bipartisan vote in the Senate died when the 112th Congress adjourned Wednesday.
"This was the most callous action I've ever seen. The leadership of this House should be condemned for it," Rep. Jerrold Nadler, D-N.Y., said during the debate.
New York Gov. Andrew Cuomo, a Democrat, said he won't trust Congress to vote as promised on the rest of the package until lawmakers show him the money.
Sandy hit the Northeast Oct. 29, causing severe damage in New York and New Jersey. U.S. damage is estimated at more than $63 billion. The storm's U.S. death toll was at least 131.
"I've heard a lot of things, and until I see the cash, I reserve judgment," Cuomo said during a meeting of his Cabinet in Albany, N.Y., Thursday.
New Jersey Gov. Chris Christie, a Republican, during a news conference Wednesday, placed the blame squarely at the feet of Boehner and the House GOP.