WASHINGTON — The sweeping tax overhaul approved by Congress this week hands Republicans a long-sought achievement they believe will bolster their defenses in next year’s midterm campaign, but party officials concede the measure may only mitigate their losses in what is shaping up to be a punishing election year.

While the tax legislation is broadly unpopular as it reaches President Trump’s desk, the bill offers Republicans the sort of signature accomplishment they have been lacking to galvanize their demoralized donors and many of their voters.

Republican lawmakers, who spent much of this year forced to explain or defend Mr. Trump’s erratic behavior, now have an opportunity to go on the offensive with an issue that unites their increasingly fractious party. And they hope that up-for-grabs voters will reward them should the economy keep growing while their tax bills are falling.

“Once the withholding tables change in January, voters will realize their paychecks are bigger as a result of tax reform,” said Representative Steve Stivers of Ohio, who runs the House Republican campaign arm. But, alluding to past midterm defeats for the party in power at the White House, he acknowledged that “history is against us.”