President Barack Obama said the Senate action put the ball in the court of House Republicans.
"With the Senate's vote, the House Republicans are now the only people left in Washington holding hostage the middle-class tax cuts for 98 percent of Americans and nearly every small business owner," Obama said in a statement released by the White House. "The last thing a typical middle class family can afford is a $2,200 tax hike at the beginning of next year.
"It's time for House Republicans to drop their demand for another $1 trillion giveaway to the wealthiest Americans, and give our families and small businesses the financial security and certainty that they need. Our economy isn't built from the top-down, it's built from a strong and growing middle class, and that's who we should be fighting for."
The Senate's 51-48 vote in favor of the Democrats' bill followed a 54-45 vote against the Republican measure, The New York Times reported. Sen. Susan Collins, R-Maine, was the only Republican to break ranks on the first vote, The Hill reported.
Minority Leader Mitch McConnell of Kentucky pushed for the votes in an attempt to put vulnerable Democrats on record backing Obama's tax cut plan.
McConnell dropped his threat to filibuster the plan.
"By setting these votes at a 50-vote threshold, nobody on the other side can hide behind a procedural vote while leaving their views on the actual bill itself a mystery to the people who sent them here," McConnell said.
Ending a filibuster takes 60 votes and McConnell said some lawmakers would be able to hide their true positions behind the procedural maneuver.
All of the tax cuts and a raft of other exemptions adopted in recent years are set to expire at the end of the year.
Prior to the second vote, McConnell noted the Democratic bill is doomed anyway, The Washington Post reported.
"The only reason we won't block it today is that we know it doesn't pass constitutional muster and won't become law," McConnell said, noting all tax legislation is constitutionally required to originate in the House.
"What today's votes are all about [is] showing the people who sent us here where we stand," he said.
McConnell said his party wanted votes on three measures -- the Democratic and Republican bills, plus a third on an Obama proposal some Senate Democrats oppose because of its changes to estate and dividend taxes.
Republicans indicated a split by the Democrats could hurt Obama politically, CNN reported.
"We think we should have a vote on all three proposals," McConnell said. "Show the American people what's really behind these proposals and what we stand for. If Democrats believe the president's rhetoric, they'll vote for his proposal. And he'll work to get their support."
The non-partisan Government Accountability Office said Monday last summer's political squabbling between the White House and Congress during the debt-ceiling crisis cost taxpayers at least $1.3 billion.
The figure -- stemming from increased Treasury Department borrowing costs after suspending investments and then re-instating them when the crisis ended -- is expected to rise as multiyear obligations and other outstanding costs are added later, the GAO said.