The legislation sped through the House on a 285-144 vote, allowing the Treasury to exceed the limit of $16.4 trillion through May 18.
The Senate is expected to approve the bill as early as Friday or next week, and President Barack Obama said he would sign the measure if it reaches his desk.
"I'm pleased that Speaker Boehner's House colleagues have decided to change course and pass a bill that defuses yet another fight over the debt ceiling," said Senate Majority Leader Harry Reid.
House Republicans cast the bill, known as the No Budget No Pay Act of 2013, as a way to force the Senate to draft a budget for the first time in four years. If either body fails to pass a budget, members of that body would have their paychecks put into an escrow account until a budget is passed. But lawmakers would be able to collect their entire salaries at the end of Congress - with or without a budget.
Rep. Paul Ryan, the Wisconsin lawmaker who will be responsible for drafting the budget for Republicans by April 15, said Congress has "a moral obligation" to prevent a debt crisis that he said will hit hardest at seniors and others who depend on government the most.
"This bill simply says 'Congress, do your job.' When I grew up in Wisconsin, if you had a job and you did the work, then you got paid. If you didn't do the work, you didn't get paid. It's that simple," Ryan said.
Ryan's 10-year-budget task will be eased in part by higher tax revenues resulting from the Jan. 1 expiration of a two-year payroll tax cut, and in part from an anticipated $600 billion generated by raising rates on upper incomes.
The Associated Press contributed to this report.
