The House on Thursday passed stalled legislation reopening the Department of Homeland Security, ending a record 76-day shutdown at the agency and resolving uncertainty over whether thousands of federal security workers would be paid in May.

The voice vote after a brief debate brought to a close a bitter partisan fight spurred by President Trump’s immigration crackdown and the tactics of federal immigration officers who fatally shot two U.S. citizens during immigration roundups in Minneapolis earlier this year. Negotiations between the White House and Democrats who were demanding new restrictions on the officers went nowhere, leading to an impasse that cut off funding on Feb. 14.

But it was a dispute among Republicans that has kept the department shuttered for nearly a month, and the G.O.P. had to bypass its own right flank to push through the bill.

Senate Republicans and Democrats had struck a deal on April 1 to fund everything except for the immigration enforcement agencies, vowing to approve that money separately in a bill that Democrats could not block. But the House G.O.P. declined for weeks to act on the measure, with conservatives refusing to vote for a bill that did not fund ICE and border patrol.

House leaders finally took it up on Thursday ahead of a 12-day break, and after the White House requested that the bill be passed immediately.

“It has come to this,” said Representative Mark Alford, Republican of Missouri, as he offered the legislation on the House floor. “We need, no we must, pay our D.H.S. workers.”

The legislation funds the department through Sept. 30, except for Immigration and Customs Enforcement and parts of border patrol, which received an earlier influx of money from a Republican-only law. Republicans in the House and Senate are now pushing new legislation that would pour an additional $70 billion into immigration operations through the end of Mr. Trump’s second term, using a process that would shield the measure from a Democratic filibuster.

Speaker Mike Johnson had sat on the funding legislation despite encouragement from the White House to pass it, as members of the House lashed out at their colleagues in the Senate for putting them in a bad political situation. To prevent employees from having to work without pay, the White House had shifted existing funds around to meet payroll after disruptions in security screening caused chaos at some airports. But the administration warned this week that it was running short of money to continue doing so.

With some Republicans still objecting to allowing the measure to come up, Mr. Johnson was forced to resort to a maneuver that sped it to the floor, limiting debate and requiring a two-thirds supermajority for passage.

In the end, opponents did not even insist on a recorded vote, though some, including Representative Chip Roy, Republican of Texas, could be heard shouting “No” as it passed.

Democrats said the impasse could have been resolved weeks ago.

“It has been Republicans who have been intransigent,” said Representative Rosa DeLauro of Connecticut, the senior Democrat on the Appropriations Committee. “It could have been done 76 days ago. But I will take it today.”

Republicans said House approval on Wednesday night of a budget outline that would pave the way for the $70 billion immigration enforcement bill finally cleared the way for approval of the funding plan, which passed the Senate on a bipartisan basis weeks ago.

Mr. Johnson said that approving the budget resolution was “critically important” to releasing the funding for the rest of the department, providing some assurance to House Republicans that sufficient money would be forthcoming even if Democrats continued their blockade. The House and Senate are to consider legislation providing the money when they return in mid-May and try to meet the president’s June 1 deadline for getting the legislation to his desk.

“Now that that box is checked, we’re allowed, then, to proceed and go through with the rest of it,” Mr. Johnson said.

The speaker acknowledged on Thursday that he had “trashed” the spending bill when the Senate first sent it to the House in March. He again called it “haphazardly drafted” and criticized it for zeroing out immigration enforcement operations.

“We threw a fit, and we had to,” he said.

But he did not explain why he backed down from a vow earlier this week to bring a modified bill to the floor.

Despite weeks of negotiations, the White House and Senate Democrats could never reach agreement on new limits on the tactics and conduct of immigration officers, including Democrats’ demands for a ban on agents wearing masks and a warrant requirement for some arrests.

Members of both parties have said the clash is further diminishing Congress’s power of the purse, as Republicans resort to the budget maneuver to fund significant parts of the department outside normal channels.

“It is really a bad way to do business, whether you are a Republican or Democrat,” Representative Tom Cole, the Oklahoma Republican who leads the Appropriations Committee, said. “It’s bad for Congress to give any president, even one you support, three years of funding this way.”