Marco Rubio, right, and Jeb Bush tangled during the debate. (Robyn Beck/AFP/Getty Images)

Fresh off a strong debate performance, Sen. Marco Rubio (R-Fla.) pivoted his attacks to the general election and Hillary Rodham Clinton, in an effort to move beyond the intraparty sniping that characterized Wednesday night's debate.

"Here's what I would hate more," he told CNN's Chris Cuomo on Thursday morning, "and that is to wake up on the first Wednesday of November in 2016 to the news that Hillary Clinton has been elected president of the United States."

"Because that means that the next four years are going to be like or worse than the last eight years, and our country can't afford that," he added.

Wednesday night's debate was the moment Rubio needed to emerge from the fray, but the task in his morning-after victory lap on television — appearing on six TV networks — was to appear presidential, and ready to take on Clinton in a general election debate.

[A breakout moment for Cruz and Rubio, but will GOP voters respond?]

As he did on the debate stage, Rubio criticized the media and Clinton simultaneously, accusing them of covering up lies about the 2012 attacks against U.S. facilities in Benghazi, Libya.

During the congressional hearing on Benghazi "it was revealed that Hillary Clinton knew early on and was telling her family and telling her friends that the attack on the consulate was by terrorists, al-Qaeda-like terrorists," Rubio said on CNN.

"The Clintons are nothing but masters of self-preservation," he added. "She was telling her relatives and friends early on that she clearly had doubts."

There was "never a single shred of evidence" that the attack was caused by a controversial video, an explanation that Clinton and Obama administration officials provided in the days after the attacks.

Clinton called the changing narratives after the attacks a "fog of war" moment in her testimony before the hearing. But Rubio and other Republicans have accused Clinton and the Obama administration of having political motivations.

"It furthered a political narrative that the administration had settled on" ahead of Obama's 2012 reelection contest, Rubio said. "That was a moment when true leadership would have said 'no.'"

On both "CBS This Morning" and ABC's "Good Morning America," Rubio blasted Clinton for tweeting a GIF showing her brushing off her shoulder during last week's House committee hearing about the deaths of four Americans in Benghazi.

"That was a serious hearing about a serious issue. As far as taking her on if I’m our nominee, we’re gonna be the party of the future and the Democrats will be exposed as the party of big government ideas from the past,” he said on CBS.

Rubio also took a swipe at CNBC, which televised the debate, saying that the questions submitted by the network's moderators weren't substantive enough. He said he thought the candidates would talk about issues including taxes and plans to reduce the debt.

"I thought it was a wasted opportunity and quite frankly that’s what made it unfair, not just to the candidates but to the American people," he said on CBS.

[GOP candidates tangle with one another — and CNBC — in a chaotic debate]

Although Rubio challenged the front-runner in the Democratic race as he made the rounds on morning news shows Thursday, he declined to attack his mentor and Republican presidential rival Jeb Bush, despite striking back at the former Florida governor during the debate.

"I still have tremendous admiration for him both as a person and what he did as governor of Florida," he told the "Today" show's Savannah Guthrie. "I'm not going to talk bad about Governor Jeb Bush. My campaign is not about him."

"It isn't going to change my feelings and my views about him," he added.

[Jeb Bush needed a debate touchdown. He didn’t come close.]

On "CBS This Morning," Rubio said the differences between the two will be fleshed out in terms of policy.

"I’m going to continue to tell people who I am, what I’m for. There are policy differences between us, we’ll discuss those. Americans deserve to hear those. But I’m not going to change my campaign," he said. "Jeb is my friend, I have admiration for him. I'm not running against him. I’m running for president."

Bush needed a breakout performance Wednesday night and didn't get it, in part because Rubio launched a counterattack against Bush when he tried to pin the senator about missing congressional votes.

[Winners and Losers from the third Republican presidential debate]

Despite his tactical victory in the debate tangle with Bush, Rubio made an effort to distance himself from his own comments that in the past suggested that he had given up on his day job in the Senate in favor of running for president.

"For me it's an incredible honor to serve in the U.S. Senate," he said on "Today," noting that constituent service is a part of the job that he enjoys "very much."

"We serve real people every day," he added. "And we enjoy doing that service and we're going to continue to do it."

[The third Republican debate transcript, annotated]

Bush didn't appear on television Thursday morning because he was traveling to New Hampshire for events.

Ana Navarro, a CNN commentator and longtime Bush ally, said he needs to immediately focus on improving his debate performances ahead of the next contest on Nov. 10 in Wisconsin.

"He’s got practically every other part down pat. He’s got the finances. He’s got the structure. He’s got the best team on the ground. He has got great policies that he’s been rolling out week after week. But this thing is really causing him some harm, and he has got to tackle it. He’s got to get over this idea that doing performance is in a way of selling out; it’s not, it’s part of the job description. Embrace it, go out there, have fun with it, and learn how to do it," she said.

Navarro said Bush can still improve his fortunes, but he needs to do so as soon as possible.

“I think Jeb still has time to turn this around, but he has got to do it in the next 11 days, before the next debate," she said. "He needs to hone down and just tackle this. Kill it dead, Jeb."