The Senate confirmed Federal Reserve Chairman Jerome Powell to a second four-year term that is shaping up to be every bit as trying as his first term as the central bank faces the highest inflation in 40 years.

Mr. Powell’s nomination, approved Thursday on an 80-19 bipartisan vote, has been on track for months to win bipartisan approval despite unease over inflation and aggressive interest-rate increases that the Fed has urgently commenced to cool price pressures.

President Biden said last fall he would reappoint Mr. Powell, opting for continuity as the scale of the central bank’s challenges in controlling inflation was becoming more evident. Mr. Powell, 69 years old, was tapped by President Donald Trump in 2018 to lead the central bank, six years after he won an appointment from President Barack Obama to its board of governors.

Mr. Powell, a former private-equity executive, was supported by lawmakers in both parties in an unusual show of broad political support for the central bank leader, who navigated the Fed’s rapid response to a pandemic-driven economic emergency in 2020. “Chair Powell is respected on both sides of the aisle for his steady leadership during the pandemic,” Sen. Mark Warner (D., Va.) said.

Six members who caucus with the Democrats and 13 Republicans voted against him.

Some lawmakers voting against him cited his handling of Fed policy as inflation rose. “Powell and the rest of the Fed have failed the American people. We should not reward failure,” Sen. Richard Shelby (R., Ala.) said in a statement.

Mr. Powell’s description of price increases last year as temporary and decisions to initially withdraw stimulus slowly—particularly after the Biden administration approved a $1.9 trillion spending bill—has drawn criticism from economists on both sides of the aisle.

“As part of restoring its credibility, the Fed needs to engage in some kind of after-action report that tries to analyze why they…were as wrong as they were in assessing the inflation risk and judging inflation to be transitory during 2021,” former Treasury Secretary Lawrence Summers said in an interview last week.

Mr. Powell conceded at a March news conference that, in hindsight, it would have been appropriate to withdraw stimulus earlier if the Fed had foreseen that the pandemic economy’s supply disruptions were going to last as long as they have, particularly after colliding with very strong demand last year.

“There was a very strong fiscal reaction. There was a very strong monetary policy reaction. Demand was really strong. Nobody should deny that,” Mr. Powell said at a March 21 economics conference. “But you could not drop that amount of demand into any of our models and produce this kind of inflation without supply-side constraints.”

Since the end of last year, Mr. Powell has pivoted the Fed toward rapidly removing stimulus. The Fed has raised interest rates twice this year, most recently last week by a half percentage point—the first such increase since 2000—to a range between 0.75% and 1%. Mr. Powell signaled further half-point increases are likely until the central bank is confident that inflation is set to slow.

Such a policy path makes it more likely that officials lift rates high enough to cause a recession. That is a different scenario from the more optimistic one sketched out in officials’ March policy projections, a so-called soft landing in which inflation falls but unemployment stays low and the economy keeps growing.

The war in Ukraine has complicated the Fed’s ability to achieve a soft landing because wars are often inflationary and the West’s sanctioning of Russia threatens to further aggravate commodity price increases and global supply-chain disruptions.

Consumer prices rose 6.6% in March from a year before, as measured by the Fed’s preferred gauge, the Commerce Department’s personal-consumption expenditures price index. On Wednesday, the Labor Department’s separate consumer-price index indicated that U.S. inflation edged down to an 8.3% annual rate in April but remained close to the fastest pace in four decades.

Lael Brainard has won Senate confirmation to become the Fed’s vice chairwoman.

Photo: Rod Lamkey/Zuma Press

Meanwhile, the unemployment rate in April stood at 3.6%, near a half-century low.

“The history is that every time we’ve had inflation above 4% and unemployment below 4%, we’ve had a recession in the next two years,” said Mr. Summers. “It’s definitely odds off that we will have a soft landing.”

One concern is that the surge in prices becomes intense enough or lasts long enough to change consumers’ and businesses’ inflation psychology, making those expectations self-fulfilling. If workers anticipate a robust inflation rate in a year’s time, they could seek higher wages now.

“We can’t allow a wage-price spiral to happen, and we can’t allow inflation expectations to become unanchored. It’s just something that we can’t allow,” Mr. Powell said last week.

Last fall, some progressive Democrats heavily lobbied Mr. Biden to replace Mr. Powell with someone who would adhere to his easy-money postpandemic stimulus policies while taking a tougher approach to financial regulation, particularly by using bank supervision tools to shape climate change policy.

White House advisers saw Mr. Powell as someone who could more easily secure Senate confirmation. They further credited him with providing a steady hand during the pandemic and an earlier interval in which he deflected attacks from Mr. Trump, who wanted more Fed stimulus before the pandemic.

Mr. Powell’s nomination was paired with the promotion of Fed governor Lael Brainard to serve as vice chairwoman. The Senate confirmed her to that position on April 26.

Lisa Cook will serve on the central bank’s board.

Photo: KEN CEDENO/REUTERS

Mr. Biden has been able to further put his stamp on the central bank with the confirmation earlier this week of two other economists to fill vacancies on the Fed’s Washington-based board of governors—Lisa Cook of Michigan State University and Philip Jefferson of Davidson College.

With Mr. Powell’s confirmation, Mr. Biden will have named four of six Fed governors to their current positions. Some analysts have speculated the new nominees might favor less-aggressive rate increases, but they are unlikely to slow the Fed from pursuing a faster pace of tightening so long as inflation remains well above the Fed’s 2% target.

Ms. Cook and Mr. Jefferson said at their Senate confirmation hearing that tackling high inflation should be a central-bank priority, and Fed governors are traditionally consensus-oriented.

The president also nominated Michael Barr, a law professor at the University of Michigan, to serve as the Fed’s vice chairman of bank supervision and fill a final vacancy on the seven-person board. His confirmation hearing is set for May 19.

Mr. Powell’s first four-year term as chairman expired in early February and he has been serving in an acting capacity as “chair pro tempore” since then. The confirmation process for the Fed nominations stalled in February when Democrats refused to move Mr. Biden’s picks individually and Republicans refused to vote on his initial selection for vice chairwoman of bank supervision, Sarah Bloom Raskin, who withdrew from consideration in March.

The political support that Mr. Powell cultivated proved valuable throughout his first term. He navigated a policy U-turn from raising rates to cutting them in 2019 while Mr. Trump threatened to fire the Fed leader for not providing easier monetary policy. Mr. Powell made clear in private that there were no circumstances, short of his death, under which he would voluntarily be forced from his job.

Later, he orchestrated one of the boldest economic policy responses since World War II, acting in concert with Congress and the U.S. Treasury. The Fed slashed interest rates to zero and then purchased trillions of dollars of government debt and offered to buy trillions more in loans and other assets to backstop credit markets.

At a congressional hearing in early March, Sen. John Kennedy (R., La.) lauded Mr. Powell’s fast action when the pandemic hit in March 2020. “Government shut down the private sector…. Markets are panicking. Everybody’s looking at you to calm things down,” he said. “You did.”

Write to Nick Timiraos at nick.timiraos@wsj.com