The $9.7 billion bill will prevent a flood insurance fund from running out of money by next week.
House Speaker John Boehner agreed to hold two votes on a total $60 billion request, after politicians from the hardest-hit areas spoke out.
The October storm flooded East Coast areas and killed at least 120 people.
It was the most costly natural disaster since Hurricane Katrina hit New Orleans in 2005.
Politicians from New York and New Jersey, the areas hardest-hit by Sandy, had complained that it took just 10 days for Congress to approve $50 billion in aid after Katrina.
More than 60 days have passed since Sandy made landfall on the US eastern seaboard.
Lawmakers remained angered by the delay even during debate on Friday.
“How dare you come to this floor and make people think everything is okay?” New Jersey Democratic Representative Bill Pascrell demanded of Republicans skeptical of the bill.
The House passed the measure by a 354 to 67 vote, while the US Senate approved it on Friday afternoon by unanimous consent.
“While we are pleased with this progress, today was just a down payment and it is now time to go even further and pass the final and more complete, clean disaster aid bill,” New York Governor Andrew Cuomo and New Jersey Governor Chris Christie said in a joint statement.
Superstorm victims have filed about 140,000 Sandy-related flood insurance claims, but many have not been fully paid out, US emergency officials have said.
“People are waiting to be paid,” Representative Frank LoBiondo, whose district includes Atlantic City and other coastal communities, said.
“They’re sleeping in rented rooms on cots somewhere, and they’re not happy. They want to get their lives back on track, and it’s cold outside. They see no prospect of relief.”
Republican Jeb Hensarling of Texas, who ultimately voted for the bill, said Sandy-related claims with the National Flood Insurance Program “need to be paid, and paid now”.
But Jeb Hensarling said the government programme was “beyond broke” and called for a bill to “transition to a private innovative, competitive, sustainable flood insurance market”.
Congress created the federal flood insurance programme in 1968 because few private insurers cover flood damage.
The vote came after John Boehner endured pointed criticism from both Democrats and Republicans on Tuesday for House leaders’ announcement the body would hold no further votes before the new Congress was sworn in.
The Senate passed a $60 billion package last week, but with the congressional term expiring on Wednesday, any Sandy-related aid legislation needed to be reintroduced.
New Jersey Governor Chris Christie was particularly outspoken. He said he had been repeatedly assured that the Sandy aid package would come to a vote before Thursday.
“There is no reason for me at the moment to believe anything they tell me,” Chris Christie said on Tuesday, adding Congress had shown “callous indifference” towards his state.
After John Boehner met with New York and New Jersey lawmakers on Wednesday, Congressman Peter King announced a two-part vote had been agreed.
A second vote on the rest of the Sandy aid package, including longer-term projects, will be held on January 15.
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