The demand that President Biden ease the migrant crisis threatening to overwhelm American cities came privately, from the New York governor to top White House officials. It came publicly, in angry statements from Democratic and Republican officials around the country. It came from scores of immigrant rights groups.

On Wednesday, the Biden administration relented.

In one of the largest such actions ever taken, the Department of Homeland Security said that almost a half-million immigrants in the United States who had fled Venezuela’s humanitarian crisis would be allowed to immediately apply for work authorization. By allowing them to legally earn income, the change could alleviate the costly burden of housing the refugees in major cities across the country. The migrants also will be protected from deportation for at least the next 18 months.

Administration officials say the decision was made, as required by law, because of the worsening conditions in Venezuela, not the situation in New York or other cities. But for Mr. Biden, the move is sure to inflame the already charged political debate, both inside his own party and with Republicans, about how to confront the surge of migration from South and Central America.

The situation at the border, where officials on Monday arrested 8,000 migrants — close to record highs in May — is providing ammunition to conservative Republicans who are vowing to shut down the government unless Congress agrees to new anti-immigration measures. They argue that protecting recent Venezuelan migrants from deportation will only encourage more to head north, hoping for similar treatment after they arrive.

Advocates for the policy say Venezuelans and other migrants decide to flee because they fear persecution, starvation and violence, not because of a policy change thousands of miles away in Washington. Mr. Biden singled out Venezuelans for the program because of their sheer numbers — they make up the largest mass migration in the hemisphere in decades.

But the dramatic move by Mr. Biden is evidence of the human dimensions and political power of an issue that has hounded him since he became president. How to deal with the border is at the heart of the funding debate in Congress, and is certain to be central to the debate between Mr. Biden and his Republican opponent in the 2024 campaign next year.

“The president is terribly compromising this country, has done irreparable harm to the country with the border invasion he has allowed,” Representative Bob Good, Republican of Virginia, said on CNN on Thursday.

The administration’s decision to expand Temporary Protected Status to an additional 472,000 Venezuelans in the United States is in line with previous moves by the administration to extend protections to some migrants from other countries. But it comes more than a year after Mr. Biden rejected similar pleas from immigration advocates to broaden its T.P.S. program for the Venezuelan migrants.

What changed, say people who engaged in the concerted effort to convince the administration to act, is the pressure campaign from members of the president’s own party.

“The images of migrants sleeping on the streets of New York City and strong, concerning statements from the Democratic mayor and other leaders highlighted the urgency for federal action,” said Krish Vignarajah, president of the Lutheran Immigration and Refugee Service.

Eleanor Acer, the senior director for refugee protection at Human Rights First, said that “it’s made a tremendous difference that cities and states were also making clear that these changes would help their communities.”

The T.P.S. law has been used for decades to provide limited legal status to people who have tried to escape natural disasters or political violence. It is designed to be a temporary refuge — usually 18 months — for people who cannot be sent home because the crisis there is ongoing. But presidents in both parties have regularly extended T.P.S. for certain groups, some of whom have remained in the United States for decades.

As of March 31, 2023, there are 16 countries whose citizens have been given T.P.S. designation, according to the Congressional Research Service: Afghanistan, Burma, Cameroon, El Salvador, Ethiopia, Haiti, Honduras, Nepal, Nicaragua, Somalia, South Sudan, Sudan, Syria, Ukraine, Venezuela, and Yemen.

“This redesignation and extension of Venezuela for T.P.S. is based on extraordinary and temporary conditions in Venezuela that prevent their nationals from returning in safety,” said Naree Ketudat, an spokeswoman at the Department of Homeland Security. “D.H.S. is doing everything in its power to get the migrants who are eligible working.”

Critics of the program have argued that the designations were being renewed so frequently that they no longer fit the definition of “temporary.” President Donald J. Trump sought to end T.P.S. protections for people from El Salvador, Honduras, Haiti, Nicaragua, Sudan and send them back to countries many had not seen in years. The effort was led by Stephen Miller, the architect of Mr. Trump’s immigration agenda.

On Thursday, Mr. Miller lashed out at Mr. Biden’s decision this week to expand T.P.S. for Venezuelans.

“By granting amnesty to the very illegals he’s escorting across the border, he will ensure the even more rapid emptying and relocation of the developing world into the United States,” Mr. Miller said. “It is the complete resettlement of America without the consent of the American people.”

Biden administration officials and advocates for migrants reject Mr. Miller’s philosophy, as do Democratic officials across the country. But the mayors and governors arguing for action over the last several months have been focused on a more pragmatic issue: how to pay to support the vast number of migrants settling in their communities.

The Biden administration has distributed close to $770 million in grants to localities, including about $140 million to New York, to bolster services for the migrants. The administration has asked Congress for another $600 million in supplemental funding for this year, and $800 million for next year.

But local officials have said they need more.

Gov. Kathy Hochul of New York, a Democrat, is in daily contact with top advisers at the White House about the issue, including Tom Perez, the president’s top liaison to other political leaders, according to one of the governor’s aides. On June 9, during a meeting on the issue in Washington with Jeffrey D. Zients, the White House chief of staff, Ms. Hochul spoke to Mr. Biden on the phone.

“She spoke to him and she made our set of asks, which included T.P.S. for Venezuelans back then,” said Avi Small, a spokesman for Ms. Hochul. “So you know, we’ve been doing our own inside game really hard.”

A person familiar with Ms. Hochul’s efforts said she helped arrange a letter from New York business groups; deployed top labor leaders to speak with White House aides; and enlisted Hillary Clinton, the former New York senator who has her own contacts inside the administration, to make the case.

Senator Chuck Schumer and Representative Hakeem Jeffries, both from New York and the top Democrats in their chambers, also worked the issue. In a joint statement on Thursday, the pair called the decision “a welcome step forward.”

The call for help came from across the country.

In August, Gov. Maura T. Healey of Massachusetts, a Democrat, declared a state of emergency, citing the cost of shelter for more than 20,000 people, including migrants, who were living in the state’s shelters. Gov. J.B. Pritzker of Illinois and Mayor Brandon Johnson of Chicago, also Democrats, called on the Biden administration to do more.

“Let me state this clearly,” Mr. Johnson said last month. “The city of Chicago cannot go on welcoming new arrivals safely and capably without significant support and immigration policy changes.”

Ms. Hochul and Mayor Eric Adams, a Democrat, began to escalate their calls for the Biden administration to expand T.P.S. this spring, as the migrant crisis in New York City hit a tipping point that prompted the mayor to bus migrants from the city to upstate communities, sparking a backlash.

In March, Mr. Adams released a 25-page blueprint that vaguely called on the federal government to implement “an expedited right-to-work policy for asylum seekers.” By April, as the situation worsened, his requests had gotten more specific: He began asking that the federal government re-designate and extend T.P.S. not only for Venezuela, but also a slew of other Latin American and African countries.

The mayor, increasingly frustrated by the burden of the situation on the city’s coffers and shelter system, also began sharpening his criticism of the Biden administration, calling the migrant crisis “a national problem dropped on the lap of a city.”

“The national government has turned its back on New York City,” he said in a speech in late April, as the federal government prepared to lift Title 42, the pandemic-era rule used to immediately expel migrants. “This is impacting our schools, public safety, our ability to take care of those who were already in shelters. This is impacting the entire city.”

As Mr. Adams ratcheted up his attacks on the White House (in speeches, television interviews, rallies), souring his relationship with the president, Ms. Hochul maneuvered more carefully, staying largely above the fray. Over the summer, the governor took pains to avoid overtly criticizing Mr. Biden and instead focused on pushing the White House for more help through back channels.

But Ms. Hochul sharply shifted gears in late August when, under pressure to take a more active role, she delivered a rare address in Albany in which she laid the blame of the crisis at the door of the White House, saying, “We’ve managed thus far without substantive support from Washington.”

In an interview on CNN, the governor said the president shared the news this week that his administration would expand T.P.S. for Venezuelans during a glitzy reception at the Metropolitan Museum of Art in Manhattan.

“The president tipped me off and said, ‘You’ve been heard, I understand,’” she said.