The wins, following the Democrats' takeover of the House in Tuesday's midterm vote, would give the party a 51-49 Senate majority, counting two independents who are expected to vote with Democrats.
But the leads held by their Democratic challengers were large enough to prompt news organizations and election experts to declare the prospects of Burns and Allen retaining their seats all but nonexistent. Their defeats would mean the GOP lost six Senate seats Tuesday.
Republican leadership aides privately acknowledged that their Senate majority was gone, but they declined to say it on the record in deference to Burns and Allen.
Control of both congressional chambers would give Democrats a powerful platform to advance their agenda. And for the first time during Bush's presidency, the party could make the legislative branch a meaningful counterweight to the White House.
Senate Democratic leader Harry Reid of Nevada is poised to join presumed House Speaker Nancy Pelosi (D-San Francisco) at the forefront of a newly energized national Democratic Party when the new Congress convenes in January.
"The American people have spoken clearly and decisively in favor of Democrats leading this country in a new direction," Reid said Wednesday night.
Also expected to assume prominent roles are several of the party's liberal stalwarts in the Senate. These include Sen. Carl Levin of Michigan, who as the likely chairman of the Armed Services Committee would be positioned to forcefully challenge the administration's Iraq policy, and Sen. Patrick J. Leahy of Vermont, who as the anticipated head of the Judiciary Committee would be positioned to tangle with the White House over the controversial prosecution of suspected terrorists and over appointments to the federal judicial bench.
The GOP had grabbed control of the Senate and the House with its landslide win in the 1994 midterm election. Republicans have held the House since then; Democrats regained a Senate majority for about 18 months during Bush's first term, but lost it in the 2002 election.
For much of Wednesday, it remained unclear who had won the Senate.
But by midday, virtually complete results in Montana showed Democrat Jon Tester ahead of Burns by about 3,000 votes — a margin that, under state law, would not require a recount. Political analysts in the state said the gap appeared too large for Burns to overcome as final tallies filtered in.
In Virginia, the Associated Press concluded Wednesday night that Democrat Jim Webb was the winner in the state's Senate race, a victory that would give the party a majority.
The wire service made its call after contacting election officials in all of Virginia's 134 voting localities. Webb, a former secretary of the Navy, had a lead of more than 7,000 votes when election officials began reviewing their initial tallies Wednesday.
Webb and Tester declared victory Wednesday. Their apparent wins completed a strong showing by Democrats that seemed unlikely even to many of the party's leaders. Although most were confident they would take the House, fewer believed the Senate was within reach.
Reid was quick to characterize the results as a mandate for sweeping change.
"The days of the do-nothing Congress are over," he said. "From changing course in Iraq to raising the minimum wage to fixing the healthcare crisis to making this country energy-independent, we're ready to get to work."
With control of the House and Senate, Democrats will have significantly more leverage over President Bush in the last two years of his administration.
Democratic-run Senate committees, like their counterparts in the House, will probably conduct more-aggressive oversight of the administration, especially of its handling of the war.