The Labour leader said hospital waiting lists and financial deficits were rising and asked for a guarantee that the NHS, which he described as a "jewel in the crown", would be fully funded.
The PM said the NHS was treating more people and outcomes were improving.
He insisted that a strong economy was the backbone of a strong NHS.
The exchange comes amid a bitter row between health secretary Jeremy Hunt and junior doctors over proposed changes to their contracts.
Mr Corbyn said the prime minister was "trying to dig himself out of a hole" over the issue and Labour would be demanding more details of the government's latest pay offer.
Turning to the wider situation in the NHS, the first time he has specifically raised the health service at PMQs since he became leader, Mr Corbyn quoted a senior figure from the Royal College of Emergency Medicine as saying "this winter will be worse than last winter and the last winter was the worst we have ever had".
Mr Corbyn said that, since 2010, waiting lists at English hospitals had gone up by a third, while 3.5 million people were now waiting to be treated. He went on to quote the King's Fund think tank which he said had warned of a "rapid decline in patient care" without extra funding.
"The NHS is in a problem, a problem of deficits in many hospitals, it's in a problem of waiting lists, it's in a problem of the financial crisis that has been faced with so many others.
"Can you address that issue and ensure everyone in this country can rely on the NHS which is surely the jewel in all of our crowns."
Mr Cameron said the Royal College of Emergency Medicine supported reforms to the NHS to create a truly seven-day-a week service and insisted that the government was committed to providing the billions in extra funding a year sought by senior NHS officials.
"I believe the NHS has the resources it needs and that is why we are seeing it treat more patients, more drugs being delivered, more tests being carried out. It is a much stronger NHS and the reason is simple. We have a much stronger economy."
For the fifth PMQs running, Mr Corbyn also raised the issue of tax credits, pressing the prime minister again for a guarantee that no-one would lose out from cuts to working tax credits from next April.
He cited the case of a serving soldier with children who he said would be Â£2,000 worse off.
Mr Cameron said the soldier in question would benefit from the increase in the personal tax allowance and an expansion in subsidised childcare but went on to defend the government's goal of a "low welfare, low tax, high pay" economy.
He told the Labour leader would have to wait three weeks until the Autumn Statement to learn the government's revised proposals following its recent defeat in the House of Commons, accusing him of opposing every single reform in the area and supporting unlimited benefits.
Mr Cameron tried to have some fun with some of Mr Corbyn's internal appointments, claiming that he had appointed a "Stalinist, Trotskyite and Communist" to his top team of advisers and he deserved "full Marx" for moving Labour to the left.
The Labour leader did not rise to the bait and stuck closely on policy issues rather than political banter.
Amid barracking from the Tory benches, he paused his questions on several occasions, telling the chamber at one point that the weekly session was not "about entertainment" but getting answers to important questions from members of the public.
The SNP's John Nicolson asked Mr Cameron whether talks to "imperil" Channel 4 by privatising it had taken place. The PM replied that he was a "huge fan" of the channel, saying all options - including private investment - were being considered to safeguard its future. The SNP's Westminster leader Angus Robertson asked about the military covenant, which the government has enshrined in law, saying many service widows were being denied forces pensions because their personal circumstances had changed. Mr Cameron said people who had remarried were now able to get the full pension and that if further steps were needed these would be looked at.
After the end of the session, which lasted nearly 40 minutes - almost 10 minutes longer than normal -  the prime minister was heard to say to the Speaker as he left the chamber that PMQs was getting "longer and longer".
Last week's exchanges finished at 12:38.
Jeremy Corbyn has warned of the risk of another "winter crisis" in the NHS as he and David Cameron clashed over health at Prime Minister's Questions.