The Senate early Wednesday passed the Republicans’ $1.5 trillion tax-cut plan, putting it closer to President Trump’s desk with an additional House vote expected Wednesday.

Senate Democrats managed to strike several provisions from a House-passed bill on procedural grounds, but Republicans nevertheless expect to send a final measure to Mr. Trump shortly to fulfill his top legislative priority and complete the most significant rewrite of the tax code in three decades.

“Many of us in this body have been waiting years for this opportunity, and millions of Americans outside of this body have been waiting even longer,” said Sen. Orrin G. Hatch, Utah Republican and chairman of the Senate Finance Committee.

The Senate passed the slightly modified plan on a 51-48 vote after the House passed its bill earlier in the day on a 227-203 vote. The procedural hiccup is not expected to significantly delay the proceedings, as the House is expected to take up the Senate-passed version Wednesday.

The action will complete a once-unthinkable timeline to get a tax bill through Congress before Christmas, and comes less than two months after House Republicans released their initial legislative text of the tax bill.

The final package cuts the corporate tax rate from 35 percent to 21 percent. It trims individual rates and lowers the top rate from 39.6 percent to 37 percent, but makes the individual cuts temporary in order to comply with Senate budget rules that imposed a $1.5 trillion cost limit.

On average, taxpayers across all income levels will see an income tax cut next year, but thanks to the phaseouts would start to see an increase long-term, according to congressional scorekeepers.

Republicans say they anticipate that future Congresses will simply extend the individual cuts.

The plan generally maintains the current progressivity of the tax code, with the top 20 percent of taxpayers paying 65 percent of the taxes next year and the top 5 percent accounting for 50 percent of the taxes.

It winds down various loopholes and exemptions while expanding others, including the child tax credit, which was a change demanded by Republican Sens. Marco Rubio of Florida and Mike Lee of Utah.

The package also repeals Obamacare’s individual mandate — partially making good on the GOP’s repeated pledges to repeal the law — and opens up the Arctic National Wildlife Refuge (ANWR) for oil drilling, which is a long-sought goal of Alaska lawmakers and many other Republicans.

Republicans say they’re moving quickly because they want taxpayers to feel the changes in their paychecks as early as February.

But there’s also a numbers issue, as the GOP ranks in the Senate will drop from 52 members to 51 once Sen.-elect Doug Jones of Alabama is sworn in, which would narrow an already tight window for passage there.

Republicans also say they need to deliver something to the voters who gave them full control of Washington, D.C., and do so ahead of the 2018 midterm elections, particularly after they failed to repeal Obamacare in its entirety this year.

Speaker Paul D. Ryan on Tuesday waved aside polling that has shown the plan isn’t all that popular, saying that public distortions might be dragging down the numbers.

“When you have a slingfest, a mudfest on TV, when pundits are slamming each other about this tax bill before it passes, that’s what’s going to happen,” Mr. Ryan said. “Results are going to make this popular.”

Democrats, meanwhile, have cast the plan as a giveaway to the wealthy, and say Republicans are rushing the process because they don’t want Americans to actually know what’s in the bill.

“This is the worst bill to ever come to the floor of the House, with stiff competition for what some of the things they’ve tried to do,” said House Minority Leader Nancy Pelosi. “The worst bill in history.”

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