With the vote held open for far longer than usual, Senate GOP leader Mitch McConnell of Kentucky and top deputy John Cornyn of Texas eventually provided the final two Republicans votes needed to overcome the Cruz filibuster bid.
Other GOP senators then changed their votes to provide political cover for McConnell and Cornyn, and all 12 Republican senators who helped Democrats stave off the filibuster then opposed the debt ceiling plan on the final vote. Needing only a simple majority for final approval, it passed with purely Democratic support.
Cruz railed against that kind of politicking in the interview Thursday with Levin, accusing fellow Republicans of being dishonest.
"The single thing that Republican politicians hate and fear the most ... is when they're forced to tell the truth. It makes their head explode," he said, calling the debt-ceiling vote "the perfect example."
His GOP colleagues, which he called establishment Republicans, wanted "a perfect show vote" in which they opposed the debt ceiling plan without being held accountable for enabling Democrats to pass it by helping them overcome the filibuster bid, Cruz said.
"They wanted to be able to tell what they view as their foolish gullible constituents back home they didn't do it, and they're mad because by (my) refusing to consent to that, they had to come out in the open and admit to that," he said.
It was unusually harsh and direct criticism of party colleagues, the opposite of the political mantra often cited by GOP icon Ronald Reagan that "thou shalt not speak ill of another Republican."
Republican anger at Cruz for the debt-ceiling maneuver had been obvious but muted.
Sen. Lindsey Graham of South Carolina said his colleagues thought "this wasn't a good strategy in light of the fact that the House showed it didn't have the ability to pass a debt ceiling with Republican votes."
Graham and other veteran GOP legislators, including McConnell and Cornyn, face primary challengers from the right this year, and the debt-ceiling vote provided ammunition for opponents to attack their conservative credentials.
For Cruz, bucking what he calls the Washington establishment, including Republicans, is all that matters.
"If we wait on the entrenched politicians in Washington to make the case to stand up, hell will freeze over before that happens," he told Levin. "This is nothing new. That's true of entrenched power. It's always been true of Washington."
About his GOP detractors, Cruz said that "every one of those senators who's angry, when they go back home, they tell their constituents they're doing everything they can to stop (the rising debt ceiling), but they don't actually want to do what they're saying."
Such a public rift with his party can hurt Republicans and Cruz, said Marc Lamont Hill, a CNN political commentator who hosts "Huff Post Live."
"Ted Cruz seems to be uncontrollable by the Republican Party," Hill noted, adding that polls show Cruz "moving in the wrong direction" among potential GOP presidential contenders in 2016. "People are dissatisfied with folk like Ted Cruz, who don't seem viable anymore because they're not willing to play ball."
Madden, the GOP strategist, conceded that with Cruz's ability to "drive a lot of headlines and get a lot of attention," Republican leaders will continue to see him "as a challenge."