WASHINGTON — President Obama and newly empowered Republican leaders professed a desire Wednesday to work together but yielded little ground on deep policy differences, foreshadowing the profound challenge of turning around a flagging economy under a divided government.

After what Mr. Obama described as an electoral “shellacking” for his party, the two sides gingerly explored the reshaped political terrain and sought to define Tuesday’s results. Republicans claimed a mandate to reverse Mr. Obama’s agenda while the president cast the vote as a cry of frustration that he has not moved fast enough.

“Over the last two years, we’ve made progress,” Mr. Obama said at a White House news conference intended to reassert his leadership as Republicans celebrated their capture of the House and gains in the Senate. “But, clearly, too many Americans haven’t felt that progress yet, and they told us that yesterday. And as president, I take responsibility for that.”

More conciliatory than contrite, Mr. Obama used that phrase, “take responsibility,” six times but rejected the suggestion that his policies were moving the country in the wrong direction. He conceded that legislation to limit greenhouse gases was dead and said he was “absolutely” willing to negotiate over the extension of tax cuts, including for the wealthy. But he drew the line at any major retreat from signature priorities, saying he would agree to “tweak” his health care program, not “relitigate arguments” over its central elements.

While Republicans also called for more cooperation, they suggested that Democrats might not have fully absorbed the lessons of their drubbing.

“Their view is that we haven’t cooperated enough,” said Senator Mitch McConnell of Kentucky, the Republican minority leader. “I think what the American people were saying yesterday is that they appreciated us saying no to the things that the American people indicated they were not in favor of.”

The trials awaiting a fractured capital could arrive swiftly when the departing Democratic-controlled Congress returns in lame-duck session this month with contentious issues like tax cuts, the federal debt limit, unemployment insurance, an arms control treaty with Russia and gay men and lesbians in the military all on the table.

As Washington awoke to the new order on Wednesday, Republicans had picked up at least 60 seats in the House, with 11 races undecided, the biggest swing since the 1948 elections under President Harry S. Truman. They took at least six seats in the Senate, falling short of control, with two races undecided.

In Colorado, Senator Michael Bennet, the Democrat, won, while in Washington Senator Patty Murray led her Republican challenger by one percentage point. In Alaska, Senator Lisa Murkowski, who ran as a write-in after losing the Republican primary, appeared poised to surpass both party nominees. If the incumbents hang onto their seats, The Democratic caucus will have a majority of 53 to 47.

The election results immediately played out on Capitol Hill as House Republicans began a leadership shuffle and Democrats awaited a decision by Speaker Nancy Pelosi of California on whether she intended to remain as her party’s leader in the minority. Ms. Pelosi told Diane Sawyer of ABC News that she would talk with her family “and pray over it” before deciding but added that she had “no regrets” and blamed the economy for her party’s losses.

“Nine and a half percent unemployment is a very eclipsing event,” she said. “If people don’t have a job, they’re not too interested in how you intend for them to have a job. They want to see results.”

Their rise to power means Republicans have more leadership positions to fill. With Representative John A. Boehner of Ohio slated to become speaker and Representative Eric Cantor of Virginia expected to become majority leader, Representative Kevin McCarthy of California, who was active in recruiting candidates this year, announced he would seek the No. 3 job of majority whip.

Representative Jeb Hensarling of Texas, formerly leader of a bloc of House conservatives, is seeking the No. 4 slot, conference chairman. He could face a challenge from Representative Michele Bachmann of Minnesota, a Tea Party favorite.

Making his debut as speaker in waiting, Mr. Boehner predicted that he would be able to work well with the incoming conservative class elected on Tuesday. “What unites us as Republicans will be the agenda of the American people,” he said. “And if we’re listening to the American people, I don’t see any problems incorporating members of the Tea Party, along with our party, in a quest that’s really the same.”

Mr. Boehner could find that unity tested, probably early next year, when the House must vote on raising the federal debt ceiling. Most Republicans in recent years have refused to support such increases, and many candidates this year ran on a platform opposing any increase in red ink. But as the party soon to be in charge of the House, Republicans run the risk of triggering a government default and a financial crisis should they refuse to increase federal borrowing power.

Mr. Boehner had no ready answer for how Republicans would handle the potentially explosive issue. “We’ll be working that out over the next couple of months,” he said.

Except for early in President George W. Bush’s tenure, when a party switch briefly handed control of the Senate to Democrats, this will be the first time Congress has been split between the parties since the 1986 election. The Senate may prove useful to Mr. Obama in killing Republican initiatives he opposes but it remains unclear whether he will be able to play off a Republican House heading into 2012 the way President Bill Clinton used a Republican-controlled Congress as a foil for his re-election in 1996.

The divide between the two chambers was evident as Senator Harry Reid of Nevada, the Democratic leader, having survived an election scare, emerged to argue that the lesson of the election was that voters want more cooperation from the parties. The onus, he said, is on Republicans.

“Republicans must take their responsibility to solve the problems of ordinary Americans,” Mr. Reid said in a conference call with reporters. “No is not the answer. It has to be yes. Not our yes, but a combined yes, something we work out, a consensus yes. The time for politics is over.”

Weakened by the election results, Mr. Obama sought Wednesday to occupy the public stage and take his punishment without surrendering stature. He announced no staff shuffle or new direction, as presidents sometimes do when they get in trouble. But he called the defeat “humbling” and said “it feels bad” to see so many allies go down for voting for his program.

“This is something that I think every president needs to go through,” he said. “In the rush of activity, sometimes we lose track of, you know, the ways that we connected with voters that got us here in the first place.”

Living in the White House, he said, “it is hard not to seem removed.”

But he quickly added, to laughter: “Now, I’m not recommending for every future president that they take a shellacking like I did last night. I’m sure there are easier ways to learn these lessons.”

Still, his analysis of that shellacking differed sharply from that of the Republicans and many independent strategists. He agreed that many voters felt government was growing too large and intrusive. But he maintained his were still the right policies.

“It would be hard to argue that we’re going backwards,” he said. “I think what you can argue is we’re stuck in neutral.”

Where he conceded a misstep was in failing to follow through on promises to reform the way Washington works out of a need to confront the economic crises he inherited: “We were in such a hurry to get things done that we didn’t change how things got done. And I think that frustrated people.”

Mr. Obama said he was “very eager to sit down” with Republicans and laid out “a whole bunch of areas where we can agree,” including job creation, deficit reduction, energy independence, education reform and infrastructure investment. While a carbon cap cannot pass “this year or next year or the year after,” he said, he suggested that he and Republicans could collaborate to promote natural gas, electric cars and nuclear energy.

He specifically embraced a proposal by Mr. Cantor to impose a moratorium on special Congressional spending items known as earmarks. Asked if there was anything in the Republicans’ Pledge to America campaign manifesto that he could support, he mentioned its promises to reform how Washington works.

“I do believe there is hope for civility,” he said. “I do believe there’s hope for progress.”