Republican disunity over the congressional tax showdown was exposed on Thursday when, in an extraordinary move, the party's leader in the Senate publicly broke ranks with his House colleagues.

After days of silence, Mitch McConnell lined up with Barack Obama and the Democrats to call for the safe passage of the payroll tax bill, instead of rallying behind his beleaguered colleague in the House, speaker John Boehner.

The rupture in the Republican party increases Obama's chances of securing a rare victory over the House Republicans, who have repeatedly humiliated him this year.

Boehner is blocking passage of a bill that would extend tax breaks to 160 million Americans, a measure introduced by Obama last year to help stimulate the US economy. If the bill is not passed by 31 December, American taxpayers face cuts in their pay of an average of $40 every two weeks. About 1.3 million people stand to lose unemployment benefit.

Obama, at a White House event organised to step up pressure on Boehner, described the standoff as "ridiculous", and paraded some of the 30,000 Americans who have written to the White House detailing the impact the tax rises would have on their lives. For some, the $40 is significant, meaning the loss of heating for almost half a week. For others, the impact is small but meaningful, from parents unable to take their children out for a pizza, to a man driving 200 miles a week to keep his father-in-law company in a nursing home.

Intent on wringing as much emotion as possible from the moment and portraying the Republicans as the party of the wealthy, Obama said: "It may be that there are some folks in the House who refuse to vote for this compromise because they don't think that $40 is a lot of money. But anyone who knows what it's like to stretch a budget knows that at the end of the week, or the end of the month, $40 can make all the difference in the world."

Boehner is increasingly isolated on the issue, criticised by Republicans from senator John McCain to party strategist Karl Rove, and one of the media's bastions of conservatism, the Wall Street Journal. They have all called on Boehner to back off, claiming he is hurting the Republican party and its chances of winning the White House next November.

But McConnell's statement presented an even bigger setback for the House speaker.

McConnell said working Americans "shouldn't face the uncertainty of a New Year's Day tax hike", and urged the House to pass the bill to avoid "any disruption in the payroll tax holiday".

The Senate on Saturday voted overwhelmingly for a deal – backed by McConnell and the Democratic leader in the Senate, Harry Reid – to extend tax cuts for two months to allow further negotiations on extending them through to the end of next year. Obama welcomed the deal.

But on Monday, House Republicans, in an unexpected about-turn, blocked it. Boehner, having reportedly reached a behind-the-scenes agreement last week with McConnell to support the bill, went into reverse, under pressure from House members close to the Tea Party movement.

The standoff with the House Republicans is turning into an early Christmas gift for Obama. Throughout this year, Boehner and his colleagues have left the president looking weak, and forced him to back down in a series of showdowns over issues including America's debt ceiling.

Grassroots Democrats repeatedly cite this supposed weakness as one of the reasons for their disappoinment with Obama. But if he can end the year with a political victory over Republicans in the House, it could help his chances of re-election next year.

Democratic strategists decided months ago Obama will fight the next election portraying the Republicans as obstructionist. He took up this theme on Thursday, saying he would sign the tax bill immediately on receiving it from Congress, and the only thing stopping it was what he called a "faction" inside the Republican party.

"What's happening right now is exactly why people just get so frustrated with Washington. This is it: this is exactly why people get so frustrated," Obama said.

"This isn't a typical Democratic-versus-Republican issue. This is an issue where an overwhelming number of people in both parties agree. How can we not get that done? I mean, has this place become so dysfunctional that even when people agree to things we can't do it? It doesn't make any sense not to reach a deal.

The president has delayed his holiday to Hawaii because of the standoff. His family left without him.

Boehner phoned Obama at the White House on Thursday morning, asking him to send an economics team to Congress to discuss the issue. Obama refused.