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The freshmen include several members who previously held House seats but also feature first-time officeholders. One thing that unifies them is the message they say they heard from voters back home: Get the government’s finances in order.

Mr. DeMint, who helped many of those freshmen get elected last year, said they are more accountable to the voters who sent them to Washington than to the leaders who once held great power over the chamber. In particular, the end of earmarks has denied leaders one tool they used to keep lawmakers in line.

Sen. Mike Lee, Utah Republican and one of those freshmen, said it was tough for him to compare before and after because he is new, but he said his stance was clear all along: no debt increase until Congress took steps to bind its own hands going forward by sending to the states for ratification a balanced budget amendment to the Constitution.

The debt vote itself provided an interesting contrast in strategy.

Most Democrats opposed the deal from the left, arguing it cut too deeply. GOP opponents, meanwhile, said it didn’t stop the sea of red ink.

Some political observers had predicted this week's House and Senate votes could turn out to be as toxic as the 2008 financial bailout vote, but judging by the substantial support in both chambers, that doesn’t appear to be the case.

In fact, vulnerable lawmakers did not shy away from the vote.

Of the more than five dozen potentially in-play House Republicans listed by political analyst Charlie Cook, just eight voted against the higher debt ceiling.

In the Senate, meanwhile, the class of senators up for election next year was the most supportive. Just six lawmakers out of the 33 seats up for election next year voted against the legislation: two Republicans, three Democrats and one independent. Of the 33 seats up in 2014, four Republicans and three Democrats voted “No.”

But of the lawmakers just elected for full terms last year, 13 voted against the debt increase, all were Republicans, and eight of them were freshmen. The ninth freshman Republican opponent was Sen. Dean Heller of Nevada, who was appointed to fill the seat of former Sen. John Ensign. Mr. Heller faces his first Senate re-election bid next year.