WASHINGTON — The Senate on Tuesday approved President Biden’s choice to run Customs and Border Protection, filling a key post that has oversight of one of the president’s earliest and biggest challenges: handling the historic spike in illegal crossings at the nation’s southern border.

With the 50-47 vote, Chris Magnus, the police chief in Tucson, Ariz., is set to become the first openly gay commissioner of the federal government’s largest law enforcement agency. He will also be the first confirmed leader the agency has had since 2019. Senator Susan Collins of Maine was the only Republican to vote in favor.

Known as a reform-minded police chief, Mr. Magnus faces the daunting task of winning the approval of the U.S. Border Patrol, an agency championed by former President Donald J. Trump that has long been criticized for excessive use of force and inhumane treatment of migrants.

Mr. Magnus’s confirmation is a belated win for the Biden administration as Republicans are rallying around border security as a key issue before the 2022 midterm elections and attacking Mr. Biden’s immigration policies as too soft.

Mr. Biden is also facing pressure from immigration advocates over policies that Mr. Magnus will eventually oversee: the continued use of a public health rule that gives C.B.P. officers the authority to quickly expel migrants, including those seeking asylum, and the recent court-ordered restart of a Trump-era program that forces certain asylum seekers to wait in Mexico while U.S. officials decide their cases.

A vote on Mr. Biden’s nominee to run Immigration and Customs Enforcement — Sheriff Ed Gonzalez of Harris County, Texas — is expected this week. Both Mr. Magnus and Mr. Gonzalez have been critical of Mr. Trump.

Mr. Magnus’s confirmation had been delayed under unusual circumstances. A member of the president’s party, Senator Ron Wyden of Oregon, refused to schedule a hearing before the Senate Finance Committee until the Department of Homeland Security provided specific answers to questions about the agency’s decision to send federal law enforcement officers into downtown Portland, Ore., last year during demonstrations protesting the murder of George Floyd by a Minneapolis police officer.

The Biden administration has faced a sharp increase in illegal crossings at the country’s southern border, at times overwhelming Border Patrol officials. The administration came under intense criticism for its response to thousands of Black migrants — most from Haiti and Venezuela — who crossed into Del Rio, Texas, in September. Images of Border Patrol agents on horseback appearing to corral migrants went viral, with some comparing it to the treatment of fugitive slaves.

Alejandro N. Mayorkas, the homeland security secretary, promised a swift investigation. But the investigating body, Customs and Border Protection’s Office of Professional Responsibility, could take months to reach a conclusion about whether there was misconduct in the agents’ actions.

During Mr. Magnus’s confirmation hearing in October, he promised lawmakers that he would be transparent about the findings.

“I have a long history of transparency and sharing things with the public, whatever the outcome may be, because I think this is how you sustain and build trust,” Mr. Magnus said at the time, adding that after Del Rio, “examining tactics and training is certainly appropriate.”

Conservatives are wary of Mr. Magnus’s commitment to enforcing immigration laws, since he has been critical of some of the policies during the Trump administration.

And winning the trust of the employees of both the Border Patrol, which has a powerful union, and Customs and Border Protection will not happen overnight.

Chuck Wexler, the executive director of the Police Executive Research Forum, who knows Mr. Magnus well, said he did not expect the new commissioner to make major policy changes to address the agency’s problems with transparency and accountability right away.

“Police chiefs coming into an environment like this recognize that their learning curve has to go up, and that means listening a lot before you do anything,” Mr. Wexler said.

Mr. Magnus has been able to work with unions in his previous jobs, and he served as head of the police union in the 1990s when he worked at the Police Department in Lansing, Mich.

He will be the second commissioner with a background in local law enforcement to lead Customs and Border Protection. A former Seattle police chief, R. Gil Kerlikowske, served in the position during part of the Obama administration.

Emily Cochrane contributed reporting.