WASHINGTON — Senator Shelley Moore Capito’s optimism about winning a seat in the Senate is tempered by another reality.

“There’s a real sense that we have to prove ourselves,” Ms. Capito, a Republican, said. In November, voters in West Virginia gave Ms. Capito a promotion from her House seat to one in the Senate that brings more staff members, more influence within her state and more power to help it in Washington.

Freed from re-election chores until 2020, she gladly accepts this advice from her retired Democratic predecessor Jay Rockefeller: “Don’t start campaigning right away.”

Yet Ms. Capito recognizes something else about this moment, for herself and fellow Republicans. “It carries responsibility,” she said.

The question is how much that responsibility changes the instincts, habits and calculations of a party that, when it comes to governing, has grown so deeply accustomed to saying no. The answer will determine whether the next two years bring any relief from the partisan inertia of the past four.

Sixty years ago, William F. Buckley Jr. offered a mission statement for the conservative movement as well as his new magazine, National Review: “It stands athwart history, yelling stop.” Then, the movement represented just one element of Republican politics, and not the dominant one.

That changed in the era of President Ronald Reagan, who declared that “government is the problem,” not the solution to America’s woes. Republican leaders who dared buck that formula since — the elder President George Bush by raising taxes or President George W. Bush by expanding Medicare and bailing out Wall Street — have faced increasing dissent.

Antigovernment fervor peaked in the Tea Party resistance to President Obama and his agenda, setting off a debt-limit crisis in 2011 and a government shutdown in 2013. Last week, it drove the unusually large intraparty opposition to the re-election of Speaker John A. Boehner, whom some conservatives accuse of insufficient zeal in battling the White House.

Yet Mr. Boehner survived. And along with Senator Mitch McConnell, the new majority leader, Mr. Boehner has signaled a more assertive approach to making Congress work, even with Mr. Obama.

Recent history points to possibilities for cooperation on initiatives the Republicans strongly favor that the Democrats could prove unwilling to block.

The 1986 midterm elections, for example, gave Democrats full control of Congress for Reagan’s last two years in office. The president and lawmakers enacted a plan — repealed the next year after public protests — to create catastrophic care coverage under Medicare.

The 2006 midterms gave Democrats the House and Senate for the last two years of Mr. Bush’s second term. Congress passed, and he signed, an increase in the federal minimum wage to $7.25 an hour. Ms. Capito, as a House member from one of the poorest states, joined 81 other Republicans in voting yes.

November’s elections produced a similar alignment, with party roles reversed. But the formula for successful legislation works only if Republicans push priorities that Mr. Obama can ultimately accept. Those do not include negation of signature White House achievements like the health care law.

Are there any priorities the president and Republicans can agree on? Representative Chris Van Hollen, Democrat of Maryland, is not sure. “Their agenda is an anti-Obama agenda,” he said.

Last year, House Republican leaders quashed a tax overhaul plan offered by Dave Camp, a Michigan Republican, when he was chairman of the Ways and Means Committee. But that was before Republicans increased their leverage by gaining control of the Senate.

“It could change behavior,” said Tom Daschle, the Democrat and former Senate majority leader from South Dakota.

At a minimum, Mr. Daschle expects Republican leaders to push for the orderly passage of annual spending bills. That alone would amount to a significant step in reducing Congress’s dysfunction.

Ratification of the Obama administration’s forthcoming Trans-Pacific Partnership trade pact represents the single most obvious target. Trade expansion remains a core tenet of Republican economic philosophy, notwithstanding some intraparty populist dissent.

Ms. Capito hopes her party can do more business with the White House than that. She and other Republicans support the Keystone XL oil pipeline. Mr. Obama favors various incentives for alternative energy that Republicans oppose, and he has signaled that he is open to negotiating on Keystone while threatening to veto legislation requiring him to approve it.

“I don’t think that’s bad for either one of us,” Ms. Capito said.

She knows Mr. Obama has no intention of abandoning the new carbon emission regulations that coal-producing states like West Virginia loathe, or acquiescing in the dismantling of his health care law. But she noted that a constituent who recently stopped her in the grocery store asked her simply to “do something about Obamacare.”

“She didn’t say get rid of it,” Ms. Capito said.

Congressional Republicans and the president have both embraced the idea of a corporate tax overhaul that lowers the 35 percent top rate, but they remain far apart on details.

But Mr. Obama’s call for a tax overhaul plan that increases infrastructure spending hardly sounds hostile to Ms. Capito. Another constituent at that grocery store asked her to help rural communities gain more Internet access, one of Mr. Obama’s oft-mentioned infrastructure goals.

Ms. Capito’s father, Arch Moore, who died last week, made highway improvements a centerpiece of his three terms as West Virginia’s governor. She said his approach to working with a Democratic-controlled Legislature was: If you are standing still, you are losing ground.

Trent Lott, the former Senate Republican leader from Mississippi, goes so far as to say Republican control of Congress “changes everything” in Washington. Some still nurture dreams of the elusive grand bargain in which Republicans accept higher taxes if Mr. Obama and Democrats agree to curb Medicare and Social Security spending.

Even in these sunny early days, Ms. Capito’s optimism is not that large.

“Before we run five miles,” she said, “we have to prove we can run one.”