“I think she’s going to make it,” said Marie Bush, a supporter of Ms. O’Donnell who went to her victory rally to cheer her on. “Too many people have been slinging mud at her, and she’s a survivor.”

Asked what the candidate might do to attract independents or even Democrats, Ms. Bush said, “I think people are smart enough now to know the world we are living in is going wrong and we need people like her to make it right.”

Republicans had been counting the Delaware seat, which was vacated by Vice President Joseph R. Biden Jr., as among those they believed they could use to reach a majority in the Senate. Party strategists said on Tuesday evening that they would assess the race this week, but that they would likely direct their money elsewhere  a sign that they believed that Ms. O’Donnell could not prevail in a general election. The Democratic nominee for the seat is Chris Coons, the county executive in New Castle County.

“There’s just a lot of nutty things she’s been saying that just simply don’t add up,” Karl Rove, the Republican strategist, said in a television interview on Fox News. “I’m for the Republican, but I’ve got to tell you, we were looking at eight to nine seats in the Senate. We’re now looking at seven to eight. In my opinion, this is not a race we’re going to be able to win.”

In New Hampshire, voters trickled into polling places for much of the day, with many precincts reporting average or lighter-than-expected turnout. Slow returns delayed the outcome, but just before midnight, aides to Mr. Lamontagne announced to a crowd of supporters at a Manchester restaurant that he was en route. He arrived to chants of “Ovide, Ovide,” with the Black Eyed Peas song “I Gotta Feeling” blasting, though he was not yet prepared to announce victory.

Mr. Lamontagne, 52, is a lawyer in Manchester who has French-Canadian roots and is deeply involved with the Catholic Church. He is a fiscal and social conservative who opposes same-sex marriage and abortion; Democrats have consistently labeled him as “too extreme” for New Hampshire. Over the course of the campaign, Mr. Lamontagne won straw polls at Tea Party events by large margins.