“The sequester and winning that fight — however you define what winning means — is critical for the party,” said Ralph Reed, the social conservative leader.

There are risks for Republicans in taking a hard line on the spending cuts, especially if the unemployment rate jumps and the economy slows. Democrats are highlighting estimates by the Congressional Budget Office that the fiscal cutbacks could leave the economy with 750,000 fewer jobs this year. And they are warning that while the effects will play out slowly, the cuts will eventually hit voters in noticeable ways.

“I don’t see how ‘I’m fine with the sequester’ differs from ‘I’m fine with slower growth and continued high unemployment, or longer delays at airports, fewer Head Start slots, furloughs among civilian D.O.D. folks’ and so on,” said Jared Bernstein, a former economic adviser to Vice President Joseph R. Biden Jr., referring to the Department of Defense. Mr. Bernstein, a senior fellow at the Center on Budget and Policy Priorities, a liberal research group, said that as the cuts became real, they would “deeply discredit” those who foster the idea that government spending is inherently wasteful and that there is no tangible cost to reducing it.

But to the degree that Mr. Boehner has prevailed in the first round of the current fight — the struggles over the across-the-board cuts and longer-term questions about addressing the national debt will go on for months and years — it could not have come at a better time for the morale of Republicans.

In the view of some conservative commentators, the Republican Party and the conservative movement are at one of their lowest points in years — “leaderless and nearly issueless,” as Rich Lowry, the editor of National Review, wrote last week in an opinion article for Politico that set off debate in the party.

With the party divided on so many other fronts, feelings are running especially high when it comes to holding the line on taxes. Conservatives are excoriating Gov. Bob McDonnell of Virginia, a Republican, for championing a tax increase to pay for transportation projects.