The measure moves now to the Senate. The Senate is expected to take up the measure Tuesday.
Northeast lawmakers from both parties have been pressing for the aid for more than two months.
The Oct. 29 storm is blamed for 140 deaths and pounded the Atlantic coast from North Carolina to Maine with hurricane-force winds and coastal flooding. New York, New Jersey and Connecticut were the hardest hit.
The vote was 241-180, and officials said the Senate was likely to accept the measure early next week and send it to President Barack Obama for his signature. Democrats supported the aid in large numbers, but there was substantial Republican backing, too, in the GOP-controlled House.
Earlier, conservatives failed in an attempt to offset a part of the bill's cost with across-the-board federal budget cuts. The vote was 258-162.
The emerging House measure includes about $16 billion to repair transit systems in New York and New Jersey and a similar amount for housing and other needs in the affected area. An additional $5.4 billion would go to the Federal Emergency and Management Agency for disaster relief, and $2 billion is ticketed for restoration of highways damaged or destroyed in the storm.
The Senate approved a $60 billion measure in the final days of the Congress that expired on Jan. 3, and a House vote had been expected quickly.
But House Speaker John Boehner unexpectedly postponed the vote in the final hours of the expiring Congress as he struggled to calm conservatives unhappy that the House had just approved a separate measure raising tax rates on the wealthy.
The delay drew a torrent of criticism, much of it from other Republicans.
"There's only one group to blame for the continued suffering of these innocent victims, the House majority and their speaker, John Boehner," New Jersey Gov. Chris Christie said on the day after the delay was announced. Rep. Pete King of New York added that campaign donors in the Northeast who give to Republicans "should have their head examined."
Less than two weeks later, the leadership brought legislation to the floor under ground rules designed to satisfy as many Republicans as possible while retaining support from Democrats eager to approve as much in disaster aid as possible.
The Associated Press contributed to this report.