The study says pensioners who take aspirin daily "as a health choice" should consider weaning themselves off of the drug.
However, doctors stressed that no one should come off the pills quickly, or without consulting their GP, as doing so would create an immediate risk of heart attack.
Around 40 per cent of UK pensioners take aspirin daily, with numbers evenly split between those who have already suffered a heart attack or stroke and those using it as a precaution.
It has long been known that aspirin carries an increased risk of gastro-intestinal bleeding but the new study, published in the Lancet, suggests the danger rises far more sharply with age than was thought.
The study's lead author, Professor Peter Rothwell, said blood thinners were causing around 20,000 bleeds annually, of which at least 3000 were fatal.
Although 20 per cent of potential recurrent heart attacks and strokes are prevented by aspirin, Rothwell, the director of Oxford's stroke prevention centre, warned: "There are also about 3000 excess bleeding deaths attributable to blood-thinners like aspirin across all age-groups."
All patients who have had a heart attack or stroke should still take aspirin, he stressed.
But those above the age of 75 should also be prescribed a proton pump inhibitor which would reduce bleeding risks by up to 90 per cent.
Those without such medical histories should consider coming off aspirin altogether, he suggested.
The study involved 3166 patients who had a stroke or heart attack, most of whom were prescribed aspirin.
For patients under 65, the annual rate of disabling or fatal bleeding was less than 0.5 per cent, rising to 1.5 per cent in those aged 75 to 84 and nearly 2.5 per cent for patients aged 85 or older. Over the decade, those over 75 had six times the risk of fatal bleeds, and a tenfold increased risk of bleeds which were either deadly or disabling.
The proportion of survivors experiencing a new or worse disability rose from 3 per cent for those under 75 to 25 per cent among older patients.
Those without a history of heart attack or stroke should avoid taking aspirin as they got older, said Rothwell.
While the drugs had "tiny benefits" for such patients in their fifties and sixties, the dangers increase with age.
"You would probably be advised to stop it in your late sixties or around 70 because at that point the risks may well outweigh the benefits," he said.
Doctors stressed no one should come off their drugs quickly or without consulting a GP.
But Dr Tim Chico, consultant cardiologist at the University of Sheffield, said the risks of aspirin were often underestimated.
"Although bleeding is a well-recognised side effect, this drug is still seen by many people as harmless, perhaps because of how easily it can be bought," he said.
