Hillary Clinton swept major primaries in Florida, North Carolina and Ohio on Tuesday, rebounding from her upset loss to Bernie Sanders in Michigan a week earlier and securing a political and psychological victory for her campaign.

The results were a significant setback for Mr. Sanders, who was counting on his fiery arguments against free trade to help him prevail across the industrial Midwest. He spent heavily trying to win Ohio, as well as Tuesday’s contests in Illinois and Missouri, but he came away with his presidential bid looking increasingly hopeless, since Mrs. Clinton is far ahead in amassing delegates needed to win the nomination.

Early Wednesday, The Associated Press declared Mrs. Clinton the winner in Illinois, too. Votes were still being counted in Missouri.

For Mrs. Clinton, Tuesday’s double-digit victories netted her so many delegates that her lead over Mr. Sanders is now about three times what Barack Obama’s was over her in 2008. On a personal level, too, she and her advisers were reassured that regardless of her Michigan defeat, her political arguments about jobs and the economy had potency in states that will be major battlegrounds in November.

The top issue for Ohio Democratic primary voters was the economy, and most of them favored Mrs. Clinton. A majority of voters also said that trade with other nations takes away American jobs, and more than half of them supported Mrs. Clinton. In Michigan, Mr. Sanders captured this group by double digits.

Mrs. Clinton was bullish and beaming at her victory party in West Palm Beach, Fla., after the first three states were called in her favor. “We are moving closer to securing the Democratic Party nomination and winning this election in November,” she said to cheers from a rowdy crowd of 1,300 people.

More than in any other primary night speech, Mrs. Clinton aimed her remarks in South Florida at the leading Republican candidate, Donald J. Trump, who boasted of his own victory just miles away.

“When we hear a candidate for president call for rounding up 12 million immigrants, banning all Muslims from entering the United States, when he embraces torture, that doesn’t make him strong — it makes him wrong,” Mrs. Clinton said, as she called on all Americans to fight against “bluster and bigotry.”

Mr. Sanders, speaking at a campaign event in Arizona, which holds its contest next week, stuck to his scathing assessment of the American economic system and promised to overhaul campaign finance rules. He criticized Walmart as not paying living wages, but also repeated his creed against global trade that has particularly hit the Midwestern industrial belt.

“I say to corporate America, you want us to buy your products, start manufacturing those products here in America, not in China,” Mr. Sanders said. 

Mr. Sanders also used the evening to criticize Mrs. Clinton for having a “super PAC” and relying on large donations. “She has received money from the drug companies and the fossil fuel industry,” he said to loud hisses and boos from the crowd of 7,200 people. “She has given speeches on Wall Street for $225,000 a pop.”

Ohio was the prize that Mrs. Clinton and Mr. Sanders coveted the most, a bellwether state in American politics that would bestow not only delegates but also a new political story line for the winner: a Clinton comeback or a Sanders surge, given that he was a long-shot there until recently.

After her loss in Michigan — and Mr. Sanders’s persistent criticism of her record on global trade — Mrs. Clinton’s aides seemed exasperated that her detailed policy positions to bring jobs back to the hard-hit Midwestern states were a harder sell than what they said were the unrealistic promises offered by her opponent. They huddled to retool her economic pitch so that it resonated better in Midwestern states in hopes of competing more strongly with Mr. Sanders.

“After having lost Michigan, we came into, particularly, Ohio with a renovated plan,” said her spokeswoman, Jennifer Palmieri.

At a rally in Youngstown, Mrs. Clinton addressed union members packed onto a factory floor at M7 Technologies, reminding them of the work she had done as a senator from New York to bring jobs back to the depressed upstate areas.

 “I stood up for our companies in New York,” she said. “I will stand up for our companies in Ohio and across America,” Mrs. Clinton said. “It’s exciting for me. I am really totally committed to bringing back manufacturing.”

The Clinton campaign dispatched former President Bill Clinton to Akron and Toledo, two predominantly white working-class cities that Mr. Sanders also campaigned in. Mrs. Clinton also racked up delegates in heavily black areas of Ohio, including courting voters in Cleveland.

Mrs. Clinton’s victories in Florida and North Carolina were due in large part to minorities: black voters in both states, and Hispanics in Florida as well. The Sanders campaign had been expecting to lose Florida, given that only Democrats could participate in the primary and he tends to perform better when independents are in the mix. But Mrs. Clinton was also broadly popular in the state, which was reflected in exit polls by Edison Research.

Mrs. Clinton drew support from about seven in 10 Hispanic voters and nearly eight in 10 black voters. She was backed by a slim majority of white voters, who accounted for about half of the electorate — down from two-thirds in 2008. She was favored by voters who put a high priority on experience or electability, and she was the solid choice when voters were asked who had the better chance of defeating Mr. Trump in November.

In Ohio exit polls, Mrs. Clinton was far ahead among black voters, but she and Mr. Sanders ran closely among white voters. As in Michigan, Mr. Sanders was drawing huge support from independents and voters under age 30, while Mrs. Clinton was popular with Democrats and voters over 45.

As Ohioans streamed to the polls on Tuesday, several voters described intense feelings for their favored candidates. At the Alliance Church in Reynoldsburg, a middle-class Columbus suburb, Sediena Barry, who installs office equipment, said she decided to vote for Bernie Sanders moments before facing the horseshoe of electronic voting machines.

 “Young people were told, get an education and work hard and you’ll get ahead, and none of us are,” said Ms. Barry, 34, who complained that she is taxed at a rate higher rate than bosses who make a half-million dollars a year.

James Moore, a restaurant manager, said he voted for Mrs. Clinton “for my personal future and my family’s future.” He added: “All her ideals line up with what we want to happen.”

For both candidates, Tuesday’s primary results represented a major turning point in the race: Roughly half of the 4,765 Democratic pledged delegates have now been awarded. But their campaigns offered sharply different takes.

Clinton advisers noted that Mr. Sanders had reached the halfway mark without showing any progress at narrowing her sizable lead in delegates, leaving him only one way forward — to win a series of landslides through the June primaries and pick up significantly more delegates than Mrs. Clinton.

That situation is highly unlikely, and Clinton advisers made it clear on Tuesday that they were confident about holding the delegate lead and ready to focus toward the general election.

“We think she will be the nominee, we’d like for this process to obviously be resolved as quickly as it can, but that’s ultimately not up to us and we’re prepared for it to go on,” Ms. Palmieri said as the voting was underway.

Sanders advisers, while acknowledging they were far behind in the delegate count, said that the race was now shifting to their advantage and that they expected to win more delegates than Mrs. Clinton in the coming primaries and caucuses. They argued that Mrs. Clinton’s best states were behind her — the primaries across the South and in Texas where her popularity among blacks and Hispanics resulted in troves of delegates.

Looking ahead, Sanders advisers predicted success in states with large liberal populations, like Wisconsin and California, and those with caucuses, a format that rewards voter enthusiasm and turnout and has favored Mr. Sanders recently. Mr. Sanders has even talked about prevailing in New York, Mrs. Clinton’s home state, given the high numbers of progressive voters and working-class Democrats who share his disgust with Wall Street.

“We’re now entering a period where we think we’ll win most if not all of the contests before the April 19 primary in New York,” said Tad Devine, a senior adviser to Mr. Sanders. “Regardless of how much momentum we have after Tuesday, the calendar is shifting in our favor. Now we just have to win some major showdowns, like in New York and New Jersey, to raise questions about whether she can really win the presidency.”

Mrs. Clinton prevailed in Ohio in 2008 against Mr. Obama and referred to that victory frequently during their campaign, arguing that the Democrats’ best hope in any general election was the winner of that state’s primary.

”It’s a state that knows how to pick a president,” Mrs. Clinton said at her victory rally in Columbus in 2008. “And no candidate in recent history, Democrat or Republican, has won the White House without winning the Ohio primary.”

Mr. Obama, of course, did go on to win the general election — and Ohio — that November. But after Tuesday, Mrs. Clinton is optimistic once again that her adage will prove true.