Republican lawmakers, who are close to many of the industries that oppose the new rule, have vowed to block it during a mandated congressional review period.

With Donald J. Trump as their presumptive presidential nominee, however, the issue is fraught with risk for Republicans. Any attempt to repeal the regulation could exacerbate an already palpable split between Mr. Trump’s blue-collar supporters and the party’s establishment donors and politicians.

Paul Porter, a truck driver from Ava, Mo., who is a member of the Teamsters union and a supporter of Mr. Trump, said he already received time and a half after eight hours of work, but strongly favored the new overtime regulation. “I have friends who are managers who get taken advantage of terribly,” he said.

The administration, for its part, expressed confidence it was holding the better political hand. “I welcome a debate in Congress about wages in this country,” Thomas E. Perez, the secretary of labor, said. “It crystallizes the difference between this president and the leadership in Congress.”

Federal employment law provides two ways for most salaried workers to become eligible for overtime. The first is through a so-called duties test that essentially determines whether or not they are bona fide executives, administrators or professionals, which has historically meant spending most of their time exercising some decision-making authority. If not, they are supposed to be eligible for overtime pay. That method is open to interpretation.

The second is more of a hard-and-fast standard, setting a salary level to determine eligibility for overtime, regardless of duties. So even for employees who are legitimately managers, if their salary is below the cutoff, they must be paid overtime.

Certain categories of workers, like teachers, doctors and outside sales representatives, continue to be exempt from the regulation, though academics primarily engaged in research are not.