Mr Brown said the plan forms part of the Government's ambition to build a National Care Service, but details of how it might be funded would have to await a forthcoming white paper.
It could benefit tens of thousands of people who end up in care when, with the proper support, they could stay at home for longer, said the Prime Minister.
The Government's Personal Care at Home Bill, going through Parliament, already includes plans to extend rights to support to an estimated 280,000 elderly people in greatest need.
In a speech today to the King's Fund health thinktank in London, Mr Brown said he wanted to go beyond this and provide "reablement" and rehabilitation support to all over-65s in England.
"Our plans for a new national care service will include a clear commitment to roll out to every area of the country... reablement and rehabilitation services," the Prime Minister said.
"This means that to help with the transition back home after a hospital stay, a residential care stay, or simply a fall or accident, people in every community will have access to intensive support services, for between four and six weeks.
"And the vast majority of beneficiaries will be older people on middle incomes or with lower incomes who struggle daily with the costs and problems of living with frailty and disability.
"They manage now, often with the help of their families and communities, but to imagine them as affluent is simply wrong. The truth is they ask for so little and deserve so much."
Mr Brown said that, too often, elderly people are admitted to hospital and "linger" there because the nursing care, rehabilitation, and simple domestic support they need to get back on their feet at home are not available.
A woman with a simple infection may end up being admitted to hospital because of the absence of facilities for care to be offered for a few days at home, said the Prime Minister, adding: "Too often, that hospital admission may mean she will never see her home again, but go from hospital direct to a care home."
Mr Brown said: "The absence of care at home has consequences - often in the form of unnecessary and expensive care elsewhere.
"Consider this fact: some NHS trusts use four times as many bed-days a year than others for older people admitted repeatedly and often unnecessarily.
"And for too long in England we have tolerated a care system which sees tens of thousands of people who would prefer to be cared for in their own homes go needlessly into residential care, with all the cost and losses that entails."
Mr Brown said extending the availability of care in the home would be "at the heart of our vision for the NHS and for social care in the next Parliament".
Proposals will be put forward to extend entitlements to specialist care like chemotherapy and kidney dialysis to take place at home.
And Mr Brown also confirmed his commitment to one-to-one personalised care for all cancer patients in their own homes, which he said could benefit around 1.6 million patients over the next five years.
The PM said dedicated nursing of the kind offered by Macmillan Nurses "has the potential to transform the experience of care for the patient and their family too".
And he added: "These bold plans for reform will be part of our ambition to create an NHS which focuses far more on prevention and early intervention - keeping people healthy wherever possible.
"By better preventive care we believe we can save 10,000 lives through a greater focus on early detection of cancer and a guarantee of one-week cancer tests.
"We believe we can prevent as many as 10,000 heart attacks and strokes each year through a guarantee of health checks on the NHS for those over 40.
"Our plans for an improved offer for patients with long-term conditions will mean fewer people needing to suffer acute admissions to hospital, and many more of those who are elderly and frail avoiding unnecessary emergency care.
"If we are successful we believe we can improve millions of lives and by better management of long-term conditions - with less need for acute admission, more people managing their own conditions in their own homes, fewer emergencies - we can save more than £2 billion over the next three years, all of which can be reinvested into the NHS, into services with the personal benefits of new dedicated nursing."
Mr Brown said improving social care was vital in an ageing society. Rather than seeing the increasing proportion of elderly people as a "threat or a burden", society must value the contribution of older people and recognise the duty to offer dignity and security in old age.
He rejected arguments that a National Care Service is unaffordable at a time when the Government is seeking to reduce a record state deficit, pointing out that the NHS itself was created when Britain was struggling to recover from the Second World War.
A radically improved social care system has the potential to save large sums by cutting hospital admissions for over-65s by more than a fifth, he said.
"Our aim is personal care whenever you need it, to the standard you need it - and as quickly as you need it," said Mr Brown.
"Others reject this idea of the way forward for healthcare and would deny families this right - we are clear. We believe in offering everyone a personal guarantee, not a gamble - legally enforceable rights to protect people in an uncertain world."
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