(AP) -- Bernie Sanders breathed new life into his long-shot White House bid with a crucial win in Michigan's primary Tuesday night, chipping away at Hillary Clinton's dominance in the Democratic presidential race. Republican Donald Trump swept to victory in both Michigan and Mississippi, overcoming fierce efforts to blunt his momentum.

Even with Sanders' win, Clinton and Trump moved closer to a general election face-off. Clinton breezed to an easy victory in Mississippi, propelled by overwhelming support from black voters, and she now has more than half the delegates she needs to clinch the Democratic nomination. Trump, too, padded his lead over Texas Sen. Ted Cruz, his closest rival.

The front-runners turned their sights on November as they reveled in their wins.

"We are better than what we are being offered by the Republicans," Clinton declared.

In a nod toward the kind of traditional politics he's shunned, Trump emphasized the importance of helping Republican senators and House members get elected in the fall. Having entered Tuesday's contests facing a barrage of criticism from rival candidates and outside groups, he also delighted in overcoming the attacks.

"Every single person who has attacked me has gone down," Trump said at one of his Florida resorts. He was flanked by tables packed with his retail products, including steaks, bottled water and wine, and defended his business record more thoroughly than he outlined his policy proposals for the country.

Sanders, meanwhile, said Michigan signaled that his campaign "is strong in every part of the country, and frankly we believe our strongest areas are yet to happen."

While a handful of recent losses to Cruz have raised questions about Trump's durability, Tuesday's contests marked another lost opportunity for rivals desperate to stop his march to the nomination. Next week's winner-take-all contests in Ohio and Florida loom large as perhaps the last chance to block him short of a contested convention fight.

Ohio Gov. John Kasich was in a fight with Cruz for second place in Michigan and hoping a good showing would give him a boost heading into next week's crucial contest in his home state.

For Florida Sen. Marco Rubio, a favorite of Republican elected officials, Tuesday marked the latest in a series of disappointing nights. He emerged from Michigan and Mississippi with no new delegates.

Rubio insisted he would press on to his home state's primary in Florida next Tuesday.

"It has to happen here, and it has to happen now," Rubio told supporters during a rally in Sarasota.

If Rubio and Kasich can't win at home, the GOP primary appears set to become a two-person race between Trump and Cruz. The Texas senator is sticking close in the delegate count, and with six states in his win column he's argued he's the only candidate standing between the brash billionaire and the GOP nomination.

During a campaign stop at a North Carolina church, Cruz took on Trump for asking rally attendees to pledge their allegiance to him. He said the move struck him as "profoundly wrong" and was something "kings and queens demand" of their subjects.

Some mainstream Republicans have cast both Trump and Cruz as unelectable in a November face-off with the Democratic nominee. But they're quickly running out of options -- and candidates --to prevent one of the men from becoming the GOP standard-bearer.

Republicans were also holding contests Tuesday in Hawaii and Idaho.

The economy ranked high on the list of concerns for voters in Michigan and Mississippi. At least 8 in 10 in each party's primary said they were worried about where the American economy is heading, according to exit polls conducted by Edison Research for The Associated Press and television networks.

Among Democrats, 8 in 10 voters in both states said the country's economic system benefits the wealthy, not all Americans.

Sanders has sought to tap into that concern, energizing young people and white, blue-collar voters with his calls for breaking up Wall Street banks and making tuition free at public colleges and universities. Michigan, with big college towns and a sizeable population of working-class voters, was a good fit for him, though something of a surprise victory given that Clinton had led in polls heading into Tuesday's voting.

Still, Sanders has struggled mightily with black voters who are crucial to Democrats in the general election. In Mississippi, black voters comprised about two-thirds of the Democratic electorate and nearly 9 in 10 backed Clinton.

After Tuesday's results, Clinton has accumulated 1,134 delegates and Sanders 502, including superdelegates. Democrats need 2,383 delegates to win the nomination.

With Tuesday's wins, Trump leads the Republican field with 428 delegates, followed by Cruz with 315, Rubio with 151 and Kasich with 52. Winning the GOP nomination requires 1,237 delegates.