One of the greatest challenges of Mitt Romney's career has always been to win over a conservative wing of the Republican party that is sceptical of his moderate record as a former governor of Massachusetts, a traditionally "blue state".

But now, as the American election enters a final frenetic month of campaigning before polling day on 6 November, the right of the Republican party is happily falling in behind Romney, lauding his performance in last week's televised Denver debate.

Yet the sudden outbreak of party unity and praise for Romney comes as the challenger is moving frantically to the middle ground, eschewing the social conservatism of his party's base and hammering away on a message of "jobs, jobs, jobs".

Romney has also apologised for secretly filmed comments at a Florida fundraiser where he said he was not worried about 47% of Americans who paid no income taxes and were dependent on government. Now he says he wants the support of 100% of Americans. He has even defended parts of Barack Obama's healthcare reforms, saying he would keep elements of the plan in place.

"He is not so much inching into the middle ground as sprinting," said Professor David Cohen, a political scientist at the University of Akron. "But we see a lot of sudden enthusiasm now for Romney because they know this is their only shot to beat Obama. A lot of Republicans saw the Romney campaign as a train wreck, a lost cause. They now see hope."

There has certainly been a Republican stampede to get behind Romney since his unexpectedly easy besting of Obama in the first presidential debate, which was watched by more than 60 million Americans. Taking advantage of a seemingly listless Obama, Romney won plaudits from across the political spectrum.

"Romney won big. [He] displayed a mastery of the economy, of small business, of healthcare, of taxes," said John Zogby, head of independent research firm JZ Analytics.

The praise has been especially loud on the conservative right. Just a few weeks ago, Wall Street Journal columnist Peggy Noonan was publicly calling the Romney camp "incompetent" and calling for an intervention by party elders. Now Noonan has dramatically changed her tune.

"The real Romney is the one who can communicate. He's straight and direct and not fancy, forgivably jargony, but worried about America and sincere. That's the Romney who showed up for the debate. Stay that guy!" she gushed in her latest column.

Others have compared Romney's debate performance with Ronald Reagan's mastery of the format and declared it a historic victory. "This could be the only presidential debate ever won so definitively in the absence of some obvious and ruinous gaffe," wrote conservative columnist Charles Krauthammer, who just a few days previously had been touring the TV talk show circuit calling Romney's campaign "small ball".

Republican activists have a simple explanation for the sudden outbreak of party discipline and their previously well-disguised ardour for Romney. "The party wants to win. The debate created a buzz and Republicans want to beat Obama. Whether you are a conservative or a moderate, that is why they are rallying behind the candidate," said Steve Mitchell, a Republican pollster.

Certainly the debate appeared to reverse what had seemed almost a death spiral for the Romney campaign, which was stuck in a cycle of gaffes – from criticising the London Olympics to the 47% video – and sinking poll numbers. Indeed there are early signs that Romney's poll numbers may now get a much-needed bump. Clarus Research released figures showing Obama's national lead had dipped to just two points and Reuters-Ipsos surveys showed Romney's popularity rising on key issues including the economy and healthcare reform in the wake of Denver.

But unrestrained Republican joy is unlikely to last long. On Friday fresh employment statistics saw America's jobless rate dip to 7.8%, allowing Obama to claim for the first time that unemployment is now lower – and the number of people in jobs is higher – than when he took up office. It is also unlikely that Obama's team will be caught as flat-footed in the remaining two debates.

The Obama camp has indulged in some public soul-searching amid fierce media criticism and is certain to come out more firmly in the next debate, on 16 October in Hempstead, New York.

On the campaign trail, Obama has already started hitting a more aggressive tone, slamming Romney as someone who shifts his opinions and has not been truthful with the electorate. He has developed a new line of attack, targeting Romney's debate vow to cut government funding to public television. "Governor Romney plans to let Wall Street run riot but he wants to bring down the hammer on Sesame Street," Obama said at a rally at the end of last week.

However, the new-look moderate Romney is likely to be a less easy target. In many ways, millions of Americans are only starting to pay close attention to the election now and the Republican nomination race – when Romney painted himself as a conservative in order to get the nod – is largely forgotten. Instead, the Democrats are facing the sort of Republican they had always feared: a moderate-sounding former governor of a blue state who talks up his economic experience and ability to create jobs. "That sort of image is not only viable for Romney to pull off, it is also absolutely necessary for Romney to do so, if he wants to be president," said Cohen.

The Republican party base – from Tea Party free marketeers to evangelical social conservatives – mostly knows that too. Romney has weathered previous criticisms of being an "Etch A Sketch" candidate who tacks to whatever position the electorate wants.

So after being almost dead in the water, Romney's campaign suddenly seems to have come to life. "It is clearly not dead now. It is a race that is going to go all the way," said Mitchell.