Republicans cemented their control of the House on Wednesday after holding onto a handful of critical seats in Arizona and California and defeating incumbent Democrats in key battleground districts, handing the G.O.P. a governing trifecta in Washington to enact President-elect Donald J. Trump’s agenda.

It was not yet clear what the margin of the Republican majority in the House would be, and preliminary counts pointed to the likelihood that they would again hold only a slight edge over Democrats. Votes were still being counted in some critical contests in the West, and a few other races were still too close to call.

The resignation of Representative Matt Gaetz of Florida on Wednesday after President-elect Donald J. Trump said he would nominate him for attorney general further complicated the math for the G.O.P. But Representative Hakeem Jeffries of New York, the Democratic leader, had already conceded that his party had fallen short, and The Associated Press declared that Republicans had effectively won control of the chamber after calling an Arizona race for Representative Juan Ciscomani.

Even a slim majority amounted to an extraordinary triumph and turnabout for Republicans, who just a year ago fretted that voters would punish them for the chaos and dysfunction gripping the House under their leadership. Instead, G.O.P. leaders have said they intend to use their chokehold on power in Washington to begin quickly passing legislation, including a sweeping package of tax cuts, loosening of regulations on domestic oil and gas production, and stringent border security measures.

“Republicans in the House and Senate have a mandate,” Speaker Mike Johnson said at a news conference in front of the Capitol on Tuesday. “It was a decisive win across the nation.”

“We’re going to raise an America First banner above this place,” he said later.

As a red wave swept across the nation, with voters registering their unhappiness with the current Biden-Harris administration, many House Democrats and the party’s candidates were ultimately unable to outrun Vice President Kamala Harris’s performance.

Nowhere did the party face more of a drubbing than in Pennsylvania, where Republicans defeated Representative Matt Cartwright, who has held his Scranton-based Eighth Congressional District seat since 2013. Representative Susan Wild, who flipped her Lehigh Valley Seventh District seat in 2018, also lost.