Mr. Balderson’s victory could be undone in November, when the party will be defending its 23-seat House majority. And in the weeks since the special election, Mr. Trump turned on Mr. Kasich in a tweet, falsely describing Mr. Balderson’s razor-thin victory as a “big win” and blaming Mr. Kasich for “tamping down enthusiasm for an otherwise great candidate.”

The Republican struggle to hold onto the 12th District highlighted a number of challenges facing the party as it goes into the midterm elections, as well as tactics that may come in handy as it scrambles to defend its congressional majority against what some believe could be a “blue wave” of liberal enthusiasm.

Mr. O’Connor all but erased Hillary Clinton’s 2016 deficit in the district thanks to his strong performance in wealthy areas and a stark gap in turnout between rural areas and more populous suburban precincts in Franklin and Delaware Counties, the district’s two largest jurisdictions.

But in the end, Mr. Balderson squeaked to victory thanks to a hard-nosed Republican playbook that relied on divisive social issues like immigration and gun rights — especially resonant in rural and exurban areas — to mobilize just enough voters to get their candidate over the finish line.

They also frequently invoked the specter of Nancy Pelosi, the Democratic Party leader who could become House majority leader if Republicans lose control of the chamber. At a rally shortly before the election, Mr. Trump cast Mr. O’Connor as a dyed-in-the-wool Pelosi supporter, even though Mr. O’Connor had said he would not vote for her to remain as Democratic leader after the midterms.