The Prime Minister said employers would "pay the price" and bosses could be barred from acting as directors if they did not ensure the wage boost was passed on.
From next April, firms will have to pay all workers aged over 25 at least £7.20 an hour - up from £6.50 now.
Writing in The Times, Mr Cameron said: "The National Living Wage will be introduced in April, giving low-paid workers a £20 a week pay rise.
"By the end of the decade, it will reach at least £9 an hour. Combine that with an increase in the Personal Allowance to £12,500, and you can see the power of the modern Conservative Party's One Nation message.
"But the National Living Wage will only work if it is properly enforced. Businesses are responsible for making that happen, and today I'm announcing how we will make sure they do."
Currently a firm is fined 100% of arrears for non-payment of the minimum wage but that will double to 200% when the new living wage comes in to force. The maximum is expected to be £20,000.
Any employer found guilty of non-compliance could be prevented from becoming a company director for 15 years, Mr Cameron added.
"We've already doubled the fines for non-payment of the National Minimum Wage – and we will double them again for that and the National Living Wage," he wrote.
"We will significantly increase the enforcement budget, set up a new team in HMRC to take forward criminal prosecutions for those who deliberately don't comply, and, from this autumn, ensure that anyone found guilty will be considered for disqualification from being a company director for 15 years.
"All that will be all overseen by a new Labour Market Enforcement Director. So to unscrupulous employers who think they can get labour on the cheap, the message is clear: underpay your staff, and you will pay the price."
Mr Cameron also took a swipe at the Labour leadership contenders - and in particular frontrunner Jeremy Corbyn.
"All of them are in a race to the left, vowing to borrow, tax and spend more – all the things that failed in the last century and were rejected at the last election," he wrote.
"Listening to some of the anti-Nato, anti-American, profoundly anti-business and enterprise debates is like groundhog day.
"Labour aren't learning. They're slaves to a failed dogma that has always left working people paying the price."
He said that Labour's "obedience" to Union bosses shows that the Tories were "the true party of working people today".
"We are changing the law so that strikes can only go ahead for essential services with a 50% turnout and 40% support. We are condemning walkouts that are not the absolute last resort," he added.
Labour leadership hopeful Andy Burnham said: "David Cameron and George Osborne's so-called National Living Wage is a complete con. Millions of young people will miss out at the same time as losing their tax credits.
"They must think again. As Labour leader, I will fight for a true living wage, available to all regardless of age, linked to the real cost of living and set around £11 nationally and £12 in London."
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