Senate Majority Leader Harry Reid called on Republicans to cooperate with an expanded Democratic majority Wednesday, a day after the Democrats picked up at least one more seat in the chamber.
"It is better to dance than to fight," said Reid, D-Nevada. "It is better to work together. Everything doesn't have to be a fight."
The last votes were still being counted Wednesday in some Western states. CNN now projects Democrat Heidi Heitkamp will beat Republican Rick Berg in North Dakota's Senate race, keeping that seat in Democratic hands after the retirement of longtime Sen. Kent Conrad.
In addition, Montana Sen. Jon Tester was projected to hold off a challenge from Republican congressman Denny Rehberg to win a second term there.
Reid spoke as the outgoing Congress is under pressure to head off a looming combination of tax increases and across-the-board cuts to federal agencies slated to kick in the first week of January. His deputy, Sen. Dick Durbin, said Tuesday's results are hugely important when it comes to that "fiscal cliff."
"We need a more positive outlook to solve our problems," said Durbin, of Illinois. "A stalemate is impossible."
Republicans captured a seat held by retiring Democrat Ben Nelson in Nebraska, where tea party-backed Republican Deb Fischer beat Bob Kerrey, a former Democratic governor and senator.
But Democrats will be replacing Republicans in Massaschusetts, where Sen. Scott Brown lost his bid for a full term to former White House adviser and bailout watchdog Elizabeth Warren; and in Indiana, where GOP veteran Richard Lugar lost a Republican primary to Richard Mourdock, who was beaten by Democratic Rep. Joe Donnelly.
The national spotlight shone on Mourdock in the waning weeks of the campaign when he discussed his views on abortion during a debate. He said: "I've struggled with it myself for a long time, but I came to realize that life is that gift from God. And even when life begins in that horrible situation of rape, that it is something that God intended to happen."
"I was attacked for standing on my principles," he said.
A Democrat also replaces one of the Senate's two independents, both of whom align themselves with the Democratic caucus, when Connecticut's Pat Murphy takes over the seat now held by Sen. Joe Lieberman.
That leaves a Senate with 53 Democrats and one independent, Vermont's Bernie Sanders -- up from the 51 Democrats and two independents in the current chamber. And voters in Maine sent another independent to Washington on Tuesday, with former Gov. Angus King taking the place of retiring Republican Sen. Olympia Snowe.
King has studiously refused to say which caucus he'll join. Though both parties say they expect him to align with the Democrats, he says he'll meet with Reid and with Senate Minority Leader Mitch McConnell, R-Kentucky, before making a decision.
"I hope I can be a bit of a bridge between the two parties," King told CNN. "When I was governor, I was an independent and worked with both sides and was able to achieve quite a bit. There were times when I agreed more with the Democrats and times I agreed more with the Republicans."
Democrats also held onto seats in Virginia and Ohio and saw several firsts. Rep. Tammy Baldwin's victory in Wisconsin will make her the first openly gay senator, in addition to being the state's first female senator. Hawaii also elected its first female senator when Democratic Rep. Mazie Hirono won, CNN projects.
There will be a record number of women in the new Senate class -- at least 19, based on CNN projections and current members.
Two years ago, Republicans had every reason to believe they could take back the Senate this year, after major midterm election gains. Going into Tuesday's vote, Republicans were protecting only 10 seats, while Democrats were defending 23 -- many in narrowly divided swing states.
But Republicans fell short of winning the seats they would need to tip the balance of power in the Senate, as controversy clouded the final days of campaigning in several races.
In August, the campaign of Rep. Todd Akin nearly collapsed after the Missouri Republican's comments about "legitimate rape" and his suggestion that women could biologically prevent pregnancy if they are raped. Until then, Republicans believed Akin would defeat Democratic Sen. Claire McCaskill, who polls showed was not very popular after just one term. McCaskill defeated Akin at the polls Tuesday, CNN projected.
Key Senate race snapshots
Arizona: Rep. Jeff Flake (R) vs. Richard Carmona (D)
Final result: Rep. Jeff Flake wins
Flake, a vocal fiscal conservative, will take the seat now held by retiring GOP Sen. Jon Kyl.
Carmona, a Hispanic Vietnam veteran and surgeon general under President George W. Bush, proved to be a formidable opponent for Flake, a six-term congressman who had to fight his way through a tough August primary. Tight general election polls caused both campaigns to go negative. Among other things, Flake accused Carmona of having anger management problems, while Carmona argued Flake failed to adequately support veterans while in the House.
No Democrat has won a U.S. Senate seat in Republican-leaning Arizona since 1988.