In a speech to the Conservative conference in Manchester delivered against a backdrop of gloomy economic data, Mr Cameron acknowledged voters' anxieties about cuts, inflation and the prospects of recovery.
Speaking amid growing concern over eurozone stability and just hours after official figures downgraded UK growth in the first two quarters of 2011, he warned that the threat to the British and global economy is "as serious today as it was in 2008 when world recession loomed".
But instead of being "paralysed by gloom and fear", he said that Britain must reject pessimism and "turn this time of challenge into a time of opportunity".
Mr Cameron promised he would offer the leadership needed to "turn this ship around". And he called on people across the country to show leadership too by working hard, building up innovative businesses and contributing to their communities.
He paid tribute to householders who went out to clear the streets following this summer's riots, teachers who have set up free schools in deprived areas and GPs who have taken control of their budgets.
And he said: "Let's reject the pessimism. Let's bring on the can-do optimism. Let's summon the energy and the appetite to fight for a better future for our country, Great Britain."
Invoking the "spirit of Britain", Mr Cameron said the UK needed to show the "hard-working, pioneering, independent, creative, adaptable, optimistic, can-do" qualities that had allowed it to make an out-sized impact on the world throughout its history.
"Remember: it's not the size of the dog in the fight - it's the size of the fight in the dog," he said. "Overcoming challenge, confounding the sceptics, reinventing ourselves, this is what we do. It's called leadership.
"We have the people, we have the ideas, and now we have a Government that's freeing those people, backing those ideas.
"So let's see an optimistic future. Let's show the world some fight. Let's pull together, work together. And together lead Britain to better days."
Mr Cameron was forced into a last-minute rewrite of his speech to clarify that he was not calling on households to pay off their credit card bills - something that economists warned would seriously damage growth.
But he said dealing with debt was the only way of escaping economic peril, rejecting Labour calls for temporary tax cuts and spending to boost growth as a recipe for interest rate hikes and further crisis.
"The only way out of a debt crisis is to deal with your debts. That's why households are paying down their credit card and store card bills," he said.
"It means banks getting their books in order. And it means governments - all over the world - cutting spending and living within their means."
On a day when the Office for National Statistics downgraded its estimate for UK GDP growth between April and June from 0.2% to 0.1%, Mr Cameron acknowledged voters had yet to see signs that his austerity programme was delivering recovery.
But he insisted: "Our plan is right. Our plan will work.
"I know that you can't see it or feel it right now. But think of it like this. The new economy we're building: it's like building a house. The most important part is the part you can't see - the foundations.
"Slowly, but surely, we're laying the foundations for a stronger future. But the vital point is this; if we don't stick with it, it won't work."
Mr Cameron rejected accusations that the Government has no plan for growth, insisting: "Whatever it takes to help our businesses take on the world, we'll do it." Deficit reduction was "line one, clause one of our plan for growth" but the Government was ready to cut regulations, get credit flowing, invest and intervene to help business.
He cited firms producing Formula One cars, jumbo jet parts and information technology as the kind of businesses that would drive Britain's "new economy".
But he said Labour's suggestion that businesses could be divided into saints and sinners to be taxed and regulated differently was a simplistic "insult" to financial services.
Mr Cameron was scathing about Labour's failure to apologise for leaving a record deficit, telling cheering activists: "We must never ever let these people anywhere near our economy again ever again."
He accused a "self-righteous" Labour of preaching equality but failing the poorest in society on education, welfare, immigration and housing.
And he denounced union plans for strike action on November 30 over public sector pensions as "not fair (and) not right".
But he also took on some of his critics in the Conservative Party, defending plans to increase in spending on international aid, recognise gay marriage and reform planning laws.
While promising he would never put the countryside at risk, he insisted that a streamlined planning system - like high-speed rail and superfast broadband - was necessary to let business thrive. And he told opponents: "Take your arguments down to the job centre. We've got to get Britain back to work."
In a speech which was light on new policy, Mr Cameron promised a "new focus" on children in care and said he would triple the scale of National Citizen Service. And he said he wanted the kind of speedy justice handed out to rioters to be applied in the courts all the time.
Mr Cameron's 50-minute speech won him a standing ovation and plaudits from Cabinet colleagues.
Katja Hall, chief policy director of the CBI, said: "The Prime Minister is right that dealing with the deficit is helping to build solid foundations for UK growth, and we need to stick to 'plan A'."
But Labour's business spokesman, Chuka Umunna, said: "On the day that official figures confirmed the economy has stagnated since last autumn, David Cameron told Britain that he would continue with an austerity plan which is hurting but not working.
"This is a Prime Minister who talks a lot about leadership but, because he and his party are wedded to outdated thinking and refuse to learn the lessons of the past, he has shown the Tories are unable to lead Britain to a better future."
PA
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