Washington (CNN) Donald Trump and Hillary Clinton carved out dominant positions in their party nominating races on Super Tuesday, marching ever closer to a scorched-earth general election clash.

Trump swamped his rivals by piling up seven wins across the nation, demonstrating broad appeal for his anti-establishment movement. Clinton also had a strong night, winning seven states and showing her strength with minorities in the South.

Trump won across the conservative South in Alabama, Arkansas, Georgia, Tennessee and Virginia, but also captured more moderate Massachusetts and Vermont.

"This has been an amazing night," Trump told reporters at his Mar-a-Lago resort in Palm Beach, Florida. He vowed to be a "unifier" and to go after Clinton with a singular focus once the GOP race eventually winds up.

But Trump's GOP rivals vowed to fight on. Ted Cruz won his home state of Texas, the biggest single prize of the night, and added Oklahoma and Alaska. And Florida Sen. Marco Rubio finally landed his first win of the 2016 season in the Minnesota Republican caucuses.

Trump's victories suggested that he did not pay a significant price for a controversy that flared in recent days over his initial failure to disavow David Duke, a former Ku Klux Klan leader, during a CNN interview, and disputes over his business record and positions on immigration.

Time running out

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And time is running out for the panicking Republican establishment to deny the billionaire the nomination, amid fears his brand of volatile anti-immigrant rhetoric could cost the party not just the White House, but the Senate.

CNN projects that Trump has so far won 233 delegates on Super Tuesday , well ahead of Cruz with 188 and Rubio with 90. That gives the billionaire a total of 315 delegates in the overall race, compared to 205 for Cruz and 106 for Rubio. A total of 1,237 delegates are required to win the Republican nomination.

In the Democratic race, Clinton won seven states , building up a delegate cushion over her insurgent rival Bernie Sanders . She rode her support among African-American voters on a Southern sweep through Alabama, Arkansas, Georgia, Tennessee, Texas and Virginia, and added Massachusetts, a state Sanders had hoped to win.

"What a Super Tuesday," Clinton declared at her victory rally in Florida, taking aim at Trump by asserting that America was already great, despite his campaign mantra, and vowing to make the country "whole again."

Sanders won his own state, Vermont, along with Colorado, Minnesota and Oklahoma. And though he failed to broaden his appeal in less liberal battlegrounds, he will now look to states in the industrial Midwest such as Michigan to inflict new blows on the former secretary of state.

But Sanders has yet to find an answer for a central question of the race: How can he win the nomination of the diverse Democratic Party without demonstrating an ability to challenge Clinton's dominance of minority voters?

The Democratic race is guaranteed to go on for months, however, because the party's system of proportionally awarding delegates means no candidate is yet close to reaching the magic number of 2,383 delegates to win the nomination.

Clinton is projected so far to win 492 delegates on Super Tuesday, compared to 330 for Sanders. That gives Clinton a grand total of 1,055 delegates -- including super delegates, who are leading party officials and lawmakers who have endorsed her campaign. Sanders has 418 delegates so far in the race. The figures are likely to be updated throughout the night.

Trump did not have it all his own way on the Republican side, following predictions he could have won as many as 10 of the 11 states up for grabs.

New life for Cruz

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Cruz won new life by capturing Texas, Oklahoma and Alaska, though he fell far short of the sweep through Southern states that once formed the central rationale of his campaign.

His three victories did, however, give him a reason to carry on in the race. He pointed to those triumphs, combined with his win in the Iowa caucuses, as proof that only he can actually beat Trump. He suggested that Rubio and others "prayerfully" consider exiting the race to unite the party.

"I am the only candidate who has beaten Donald three times," Cruz told CNN's Wolf Blitzer.

And Rubio, after suffering a string of miserable election nights, finally secured his first win of the campaign in Minnesota.

He argued that Trump could not amass the 1,237 delegates needed to win the Republican nomination once winner-take-all contests begin to crop up on the calendar later this month --including his own, must-win state of Florida.

"This is the fight for the heart and soul of the Republican Party," Rubio told CNN's Jake Tapper. "I will go through all 50 states before we stop fighting to save the Republican Party from someone like that."

But his claim that he can unite the Republican Party against Trump looks increasingly questionable, given his losses to the former reality television star in other target states such as Virginia.

In some states, it was clear that Rubio and Cruz were dividing the opposition to Trump, who is still benefiting from the split field against him.

But there seems little incentive for either candidate to get out. Rubio has sufficient support and financial resources to continue and could benefit from an emerging effort by anti-Trump forces to target the billionaire with a super PAC.

The same is true of Cruz, and he and Rubio, youthful first term senators, are locked in a battle for the future leadership of the party, and don't seem likely to join together to present an anti-Trump front.

And given the fact that Cruz, who is widely disliked among his peers in Washington, and Trump have won all but one of the contests so far, it is clear the establishment is even farther away from providing a credible challenger for the nomination.

Sanders also is vowing to stay in the campaign -- and with his lucrative army of small donors and grass-roots appeal, he has no reason to leave.

"This campaign is not just about electing a president," Sanders said at a rally Tuesday night in Vermont. "It is about transforming America."