WASHINGTON — The House on Tuesday approved the most significant overhaul of the Postal Service in nearly two decades, seeking to address the beleaguered agency’s financial woes and counter pandemic-era mail delays that became a flash point in the 2020 elections.

The service has teetered on the brink of insolvency for years, as repeated efforts to revamp its structure and finances have stalled in Congress. In 2020, a slowdown of mail delivery and a series of operational changes ahead of the election, in which a record number of votes were cast by mail because of the pandemic, prompted intense scrutiny and helped set off a renewed effort to address the agency’s troubles.

The House approved the measure 342 to 92.

A companion bill in the evenly divided Senate has bipartisan backing, including more than a dozen Republican co-sponsors, signaling sufficient support for the measure to move through that chamber and become law.

“Given the significant, bipartisan support for the same bill in the Senate, I expect to move quickly to vote on these critical reforms that will help ensure the Postal Service’s long-term success,” said Senator Gary Peters, Democrat of Michigan and a champion of the companion bill, in a statement.

The Biden administration also signaled its support for the measure in a statement on Monday, saying it backs “efforts to strengthen the United States Postal Service, including by providing postal employees with the dignity, fair pay, and employer-provided benefits they have earned.”

Despite being a popular mainstay of American life, the Postal Service regularly fails to turn a profit, with 2020 marking the 14th consecutive year it incurred a net annual loss. Its revenue comes mostly from the sale of postage and other products, and without congressional action, lawmakers warned the agency could run out of cash by 2024.

During a meeting with the Postal Service’s Board of Governors on Tuesday, Louis DeJoy, the postmaster general, called the bill “long overdue legislation,” according to a transcript of the meeting. He praised lawmakers involved in the measure and expressed optimism that the Senate would soon take up the bill, which he described as “legislation that is vital to the United States Postal Service and the American people.”

To address the financial strain on the agency, the bill requires retired employees to enroll in Medicare when they are eligible and removes a mandate, first imposed by a 2006 law, that the agency cover its future health care costs decades in advance. The Postal Service estimates that those two changes will save the agency about $50 billion over a decade, according to a fact sheet provided by the House Committee on Oversight and Reform, whose leaders led efforts to draft the legislation. The committee said it would be the most significant overhaul of the agency in nearly two decades.

The measure also would require the agency to publish delivery data that customers could search using a street address, ZIP code or post office box, as well as mandating the agency maintain a delivery standard of at least six days a week. A 10-year strategic plan issued by the Postal Service in March that proposed reducing hours and lengthening delivery times drew immediate pushback from Democrats.

In the months leading up to the 2020 election, Democrats also sparred with Mr. DeJoy, a Trump megadonor, over a series of cost-cutting measures that eliminated overtime for carriers, reduced post office hours and removed postal boxes. He decided to suspend those changes until after the 2020 election after criticism that they could curtail vote by-mail efforts amid the pandemic.

The bill passed on Tuesday also would expand special rates for local newspaper distribution, and would also require regular reports to Congress about the agency’s financial state.

The legislation was led by Representative Carolyn B. Maloney, Democrat of New York and the chairwoman of the oversight committee, and its top Republican, Representative James R. Comer of Kentucky.

“It’s an American value — I wanted to do everything I could to preserve it,” Ms. Maloney said in an interview. “It’s a problem that needed to be fixed.”

In remarks on the House floor, Mr. Comer noted that the legislation is projected to save federal dollars with its reforms, adding that “this targeted bill addresses the immediate needs of the Postal Service to help it succeed into the 21st century.”

Lawmakers collected a series of letters of support from companies and outside groups to illustrate the widespread support for the legislation, including from Hallmark Cards and the Greeting Card Association.

Brian Huseman, the vice president of public policy at Amazon, said the legislation would “ensure the agency can continue to provide reliable service to the American people and serve as an excellent delivery and logistics provider in the years ahead.”