WASHINGTON — The stunning Democratic wins in two Georgia Senate races this week upended Washington’s power structure overnight, providing an unexpected opening to the incoming Biden administration by handing unified control of Congress to Democrats, who will be tested by governing with spare majorities.

The victories by Raphael Warnock and Jon Ossoff mean that Senator Chuck Schumer, Democrat of New York, will control the Senate floor rather than Senator Mitch McConnell, Republican of Kentucky and a man Democrats have long seen as the main impediment to their legislative ambitions.

The momentous shift occurred even as a violent siege of the Capitol on Wednesday, egged on by President Trump, made clear the staunch refusal of his supporters to acknowledge President-elect Joseph R. Biden Jr. as the winner of the election, an explosive last gasp of Republican protest before Democrats assume full control. Thrust together at a secure location with top congressional leaders after being evacuated during the mayhem, Mr. McConnell found himself congratulating Mr. Schumer on his newfound status.

In a wholesale change that will shift the policy agenda after Mr. Biden’s inauguration, liberals — including Senator Bernie Sanders of Vermont, the democratic socialist who will now lead the Budget Committee — will head Senate panels, rather than conservatives. Legislation from the Democratic-controlled House that had languished in the Senate will now get consideration across the Rotunda.

The abrupt shift in circumstances invigorated Democrats who had been deflated in November when they failed to gain a Senate majority on Nov. 3 despite Mr. Biden’s victory. Given the traditional advantage Republicans have had in Georgia runoff elections, many Democrats had become resigned to the prospect that they would be sentenced to another two years in the Senate minority, stymied in delivering on Mr. Biden’s priorities.

“We sure did not take the most direct path to get here, but here we are,” said Mr. Schumer, happy with the outcome any way he could get it, a result that put him in reach of fulfilling his ambition of becoming majority leader after four years as the chief of the minority.

While the change in Senate control is momentous, particularly in easing the way for Mr. Biden to fill administration jobs and judicial vacancies, it does not mean that Democrats can have their way on everything — or even most things.

The Democratic majority in the House shrank in the last election, emboldening Republicans and giving Speaker Nancy Pelosi less wiggle room in what is likely her last term. More than half of House Republicans voted to throw out certified presidential election results from Arizona and Pennsylvania overnight Wednesday and Thursday without evidence of fraud, reflecting both the extreme character of the House Republican conference and what is sure to be a reluctance to work with Mr. Biden.

With the Senate divided 50 to 50 and Democrats in charge only by virtue of the tiebreaking power of the vice president, the filibuster also looms large. Democrats will need to attract at least 10 Republicans to advance most bills while contending with demands from the left for bolder action now that their party will control all of Congress.

Democrats conceded the difficulties but still welcomed the reversal of fortune.

“It is not all going to be easy, but it is certainly better than being 52-48 and President Biden playing ‘Mother, May I?’ with Leader McConnell in moving any legislation to the floor,” said Senator Chris Coons, Democrat of Delaware, one of the incoming president’s closest allies on Capitol Hill.

Yet Mr. McConnell, newly elected to his seventh term, has been in the position of leading the minority before and has proved effective in obstructing Democratic priorities.

During President Barack Obama’s first term, Democrats had a filibuster-proof 60 votes for a period, and Mr. McConnell still managed to confound Democrats while gradually chipping away at their majority. Republicans took control in 2015, mainly through emphasizing party unity against Democratic initiatives.

As minority leader, Mr. McConnell can be expected to employ the same tactics while focusing on the 2022 midterm elections and seeking to regain his Senate power. That will make the first two years of Mr. Biden’s administration extremely important when it comes to accomplishing any major priority.

Republicans said they recognized that the legislative environment will be drastically different.

“It’s the agenda, an agenda shift — totally changed,” said Senator Shelley Moore Capito, Republican of West Virginia. “They’re going to have the ability to run things from the House and, you know, shift the emphasis.”

When the Senate last had a 50-to-50 split in 2001, the two leaders, the Republican Trent Lott of Mississippi and the Democrat Tom Daschle of South Dakota, worked out a power-sharing agreement. But those two leaders had a much deeper relationship than Mr. McConnell and Mr. Schumer — they had worked cooperatively on the impeachment trial of President Bill Clinton — and the Senate was less polarized than it is today.

Mr. Schumer and Mr. McConnell will need to engage in talks to come up with some sort of governing framework.

“I assume in the next couple weeks, Schumer and Mitch will sit down and kind of figure out how this is going to work,” said Senator John Thune of South Dakota, the No. 2 Republican. “We had a little bit of a pattern back in 2000, but times have changed. It’s different now.”

Perhaps the biggest difference will be the committee chairmen, representing a significant swing in ideology. Besides Mr. Sanders, for example, Senator Sherrod Brown, the progressive Ohio Democrat and strong labor ally, is set to be head of the Banking Committee and will have a markedly different agenda than that of the outgoing Republican chairman, Senator Michael D. Crapo of Idaho.

Mr. Brown said his first order of legislative business would be addressing the effect of the coronavirus pandemic and relief provisions set to expire, including an eviction moratorium.

“We need to fix a lot of the damage Trump’s done, and then there’s pent-up demand for a whole lot of things,” Mr. Brown said. “What do we do about climate and about racial inequality, about wealth inequality, about structural racism?”

Among other notable committee changes would be Senator Ron Wyden of Oregon as head of the tax-writing Finance Committee, and Senator Richard J. Durbin of Illinois as chairman of the Judiciary Committee rather than Senator Charles E. Grassley of Iowa, who was a chief driver of the Republican push to install more than 200 conservative judges on the nation’s federal courts the past four years. Senator Patty Murray of Washington, an aggressive backer of health law changes, is in line for the health committee.

With the even partisan split, Democrats have begun talking about employing a special legislative process called reconciliation that applies budget rules to eliminate the threat of a filibuster, but what can be accomplished with that approach is limited. Activists are encouraging Democrats to try to eliminate the 60-vote filibuster to take advantage of their power while they have it.

“A window of opportunity like this may not come around again for a long while,” said Brian Fallon, a former Schumer aide and head of the progressive group Demand Justice. “It is almost overwhelming to think of all the opportunities for legislating that now exist, but the priority must be democratic reforms that make institutions like the Senate and our courts more aligned with the will of the people.”

But a handful of centrist Democrats, including Senator Joe Manchin III of West Virginia and Jon Tester of Montana, have said they have no interest in gutting the filibuster, instead regarding it as a way to force the kind of compromise they think could restore the Senate’s ability to legislate.

“Bipartisan legislation tends to stand the test of time, and so hopefully we continue to work together and have it be encouraged by the filibuster,” Mr. Tester said.