Republicans were making early gains on Wednesday in their drive to maintain control of the House, holding a handful of critical seats that Democrats had sought to flip even as the G.O.P. lost a pair of districts in New York.

Political strategists and polling in both parties have indicated for weeks that the fight for the House majority would be exceedingly tight, and it appeared as of Wednesday afternoon that control of the chamber would be too close to call for some time.

But optimism was waning among Democrats that they could flip the House and create a beachhead of resistance to a Republican-controlled Senate and White House held by Donald J. Trump. Instead, Republicans were confidently predicting they were headed toward a governing trifecta in Washington that would give them a free hand to enact Mr. Trump’s policy agenda.

Democrats flipped two major seats in New York, defeating Representatives Marc Molinaro in the Hudson Valley and Brandon Williams in Syracuse. They held two key seats — one in Michigan, and another in New Mexico — that Republicans had hoped to pick off. And they were hopeful they could prevail in contests in Arizona and California, where votes were still being counted.

But the party appeared to collapse in Pennsylvania, in an early indication of the difficulty that even battle-tested incumbents were facing trying to outrun Vice President Kamala Harris in a state where Mr. Trump emerged triumphant. Republicans were set to oust Representative Matt Cartwright, who has held his Scranton-based seat since 2013 and has long defied political gravity in his conservative-leaning district. They were also set to flip the Lehigh Valley district that Representative Susan Wild won in 2018. Both Democrats conceded defeat on Wednesday, even before the final results were known.

And Democrats failed to make inroads in a number of critical districts that the party needed to win to wrest back control of the House, as voters nationwide registered their unhappiness with the Biden-Harris administration, setting up Mr. Trump to become the first Republican to win the national popular vote since 2004.