WASHINGTON — With sweeping victories that exceeded their own sky-high expectations, the GOP has dealt President Barack Obama and Democrats the most devastating electoral defeat of his presidency. Their prize is full control of Congress, and with it, the power to shape the direction of America’s government in the next two years.

Both parties talked Tuesday about need to compromise, but they will face tough obstacles in following through. The list is long: the already looming 2016 elections, persistent divisions within the Republican Party, and the frosty relationship between Obama and Sen. Mitch McConnell, who won re-election in Kentucky and is likely to ascend to majority leader.

“I don’t expect the president to wake up tomorrow and view the world any differently than he did this morning,” McConnell said at his victory party Tuesday night. “He knows I won’t either.”

The election puts Republicans back in power in the Senate for the first time in eight years, and alongside a GOP-led House, the party will set a legislative agenda unlike anything that would come from Obama’s White House. The president’s top advisers have spent weeks planning for how to deal with a Republican-led Senate. Obama will hold a press conference Wednesday afternoon, and plans to meet with congressional leaders Friday at the White House.

In the rosiest of scenarios, McConnell and Obama can look for common ground in areas where their parties have overlapping interests: overhauling the nation’s complicated tax code, repairing crumbling roads and bridges, and inking free-trade agreements with the European Union and Asia-Pacific nations.

“This is a new chapter in the presidency and it doesn’t have to be a defeated one,” said Bill Burton, a former Obama White House and campaign adviser. “We lost in a lot of places, but the truth is this could open up some real opportunity to actually get some things done.”

For Obama, who has grown resentful of his diminished political standing, the prospect of reaching accords with a GOP-led Congress is a consolation prize that could help salvage his flailing second term. Republican Senate leaders may also see something to gain by showing Americans they can govern effectively, given that voters expressed dissatisfaction with their party as well with Obama, according to exit polls conducted for The Associated Press and television networks.

But McConnell and Republicans are likely to have plenty on their to-do list that doesn’t match Obama’s plans for his final years in office, including cutting budget deficits, making changes to Obama’s signature health care law and approving construction of the contentious Keystone XL pipeline from Canada.

“We will send the president bill after bill until he wearies of it,” said Kentucky Republican Sen. Rand Paul, a tea party favorite and potential 2016 presidential candidate.

That fast-approaching campaign likely leaves McConnell and Obama a short window in which to make progress on any compromise legislation.

McConnell will be fending off pressure from prospective White House hopefuls, including Paul and Sen. Ted Cruz, R-Texas, who see little incentive to compromise with a Democratic White House as they appeal to the conservative voters who decide GOP primaries. Republican opposition to the president is likely to only deepen if the president accelerates his use of executive actions, including presidential directives on immigration that are expected before the end of the year.

Obama, too, will be battling internal party politics. He may no longer be able to count on full support from Democrats who are unlikely to want to help the GOP look effective in governing during the lead-up to a presidential contest. Tuesday’s elections also strip him of some of the more moderate members of his caucus, such as Sen. Mark Pryor of Arkansas and Sen. Kay Hagan of North Carolina, leaving him with more liberal members who have fewer overlapping priorities with Republicans.

The White House was silent as Republicans racked up one win after another Tuesday and won control of the Senate hours — if not days — before almost anyone expected. Privately, there was frustration among some advisers that Democrats wary of being associated with Obama largely sidelined the increasingly unpopular president throughout the campaign. That strategy did little to stem losses in places where Obama showed prowess in his runs for president, including Colorado, Iowa and Florida.

For Democrats, the one silver lining in Tuesday’s elections is that they are now over. Attention can now turn to 2016, when the Senate contests will largely take place in states that are traditionally more favorable to Democrats.

And while they may still fret about being saddled with an unpopular president at their party’s helm, many are already prepared to move on. An announcement from Hillary Rodham Clinton, the political juggernaut who appears poised to run to replace Obama, is expected around the end of the year.

Julie Pace has covered the White House for The Associated Press since 2009.