WASHINGTON — The Senate on Thursday cleared away the last major hurdle to raising the debt ceiling, effectively ending a Republican blockade and all but guaranteeing that Congress will be able to move quickly in the coming days to steer the government away from a first-ever federal default.

In a vote of 64-36, 14 Republicans joined every Democrat to take up legislation that would smooth and speed the way for Congress to increase the statutory borrowing limit by a set amount, still to be determined. The action broke through months of politically charged wrangling in the 50-50 Senate, where Republicans refused to let Democrats take up any long-term debt ceiling increase, using the filibuster to block their efforts to do so.

The stalemate ended this week, when Senator Mitch McConnell, the Kentucky Republican and minority leader, cut a deal with leading Democrats on a convoluted process that would allow the borrowing cap to be raised on a simple majority vote, with only Democrats supporting it, rather than subjecting it to the 60-vote threshold that applies to most major legislation.

Mr. McConnell was among the Republicans who voted to advance it, amid recriminations from lawmakers and activists in his party who said it was a betrayal. Its passage was expected as early as Thursday evening, sending it to President Biden for his signature.

That would pave the way for a separate vote to raise the limit by as much as $2.5 trillion, expected early next week. It puts Congress on track to avoid a fiscal crisis with little time to spare. The Treasury Department has said that it could breach the statutory debt limit soon after Dec. 15, and would no longer be able to finance the government’s obligations.

The measure was packaged with legislation that would postpone scheduled cuts to Medicare, farm aid and other mandatory spending programs, a sweetener for reluctant Republicans who have held firm against giving Democrats the ability to raise the debt ceiling. It would establish a one-time, fast-track pathway for the Senate to raise the debt limit by a specific amount with a simple majority vote, clearing away the threat of a filibuster or other procedural hurdles that would allow Republicans to stop its passage.

“The nation’s debt has been incurred on a bipartisan basis, so I’m pleased that this responsible action will be taken today to facilitate a process that avoids a default,” said Senator Chuck Schumer, the New York Democrat and majority leader. He thanked Mr. McConnell for the “fruitful, candid, productive” discussions that led to the agreement.

“I’m optimistic that after today’s vote, we’ll be on a glide path to avoid a catastrophic default,” Mr. Schumer added.

While Democrats have not said how much they will increase the borrowing cap, it is expected that they will try to delay another fiscal cliff until after the midterm elections next year. One Treasury estimate suggested they would need to raise it by as much as $2.5 trillion to cover that period, according to a person familiar with the preliminary accounting who disclosed it on the condition of anonymity.

The legislative contortions were necessary because of Republicans’ intransigence on the debt limit. Given that Democrats are using the fast-track budget reconciliation process to muscle through Mr. Biden’s $2.2 trillion climate, tax and social spending bill over their opposition, Republicans had demanded that Democrats use the same maneuver — which shields legislation from a filibuster — to address the debt limit.

Democrats objected, arguing that both parties were responsible for raising the borrowing cap to accommodate spending that had been approved by and incurred under both Republican and Democratic administrations. Reconciliation, they added, would be an unnecessarily complex and time-consuming way to do so.

In October, Mr. McConnell relented temporarily, corralling 10 of his colleagues to join him in breaking his own party’s filibuster of a short-term increase to the debt limit, which then passed with only Democratic votes. But he warned in a scathing letter to Mr. Biden that he would not do so again.

A month later, Mr. McConnell and Mr. Schumer began quietly discussing alternatives, including the possibility of attaching a debt-limit increase to the annual defense policy bill, the last must-pass piece of legislation trudging through Congress before the end of the year.

On Tuesday, congressional leaders announced the deal and the House passed the measure later that day with all but one Republican voting in opposition.

Several Republicans seethed over the agreement, accusing Mr. McConnell and his allies of caving and warning that the maneuver would set up a dangerous precedent that could erode the rules of the Senate filibuster that have long protected the minority party.

“It sure as hell sets in motion playing with the rules of the Senate in a fashion that I never even thought of until 24 hours ago,” said Senator Lindsey Graham, Republican of South Carolina. “I want to make this hard, not easy, because I think what we’re doing is going to really change the structure of the Senate and certainly going to do a lot of damage to the Republican Party.”

But as he did in October, Mr. McConnell was able to scrounge together support from a coalition that included members of his leadership team, centrists and retiring lawmakers. Republicans said that as long as Democrats took responsibility for the final vote raising the limit, they would be satisfied, promising to use the vote as a political cudgel in the midterm elections.

A few Republicans were also swayed by the decision to package the debt limit process with the postponement of mandatory cuts to a range of federal spending programs, including community block grants, farm aid and a 4 percent reduction in Medicare payments to doctors and hospitals. The legislation would also extend a temporary pay increase in Medicare implemented earlier in the pandemic, allowing doctors and hospitals to keep their current pay rates until April before the cut begins to phase out.