The HPA poll of 1,800 people in England also found one in 10 people keep leftover antibiotics - and many would self-medicate next time they got ill.
A leading GP said antibiotics were not a "cure all".
The HPA's Dr Cliodna McNulty said self-medicating was unsafe and could fuel drug resistance.
Dr McNulty, head of primary care for the HPA, said: "The majority of people can treat themselves at home using over-the-counter medicines to relieve symptoms."
Of those polled, 500 had been prescribed antibiotics in the previous year, with 11% reporting they had leftovers and 6% saying they might take them if they had future infections.
Speaking on European Antibiotics Awareness Day, Dr McNulty said that while the numbers might appear small, they could translate into large numbers given that 30% of people take antibiotics every year.
She said: "There is evidence that the more antibiotics you have, the more likely you are to develop resistance. And you're also more likely to develop antibiotic-related diarrhoea."
But 70% were aware of the problem of antibiotic resistance in hospitals, and a similar number were aware that they or their family could be affected.
'Not a cure-all' The HPA says health professionals must learn to resist demands from patients for treatments they know have little or no effect on coughs and colds. It found 97% of those questioned said that the last time they had asked their GP or nurse for an antibiotic, they were prescribed one.
TOP TIPS FROM THE HPA
- Most coughs and colds get better on their own - antibiotics will not speed recovery
- Talk to your GP about whether you need them
- Coughing up phlegm on its own is not a reason to need an antibiotic - even if it is yellow
- A sore throat plus runny nose with phlegm suggests the infection is less likely to respond to antibiotics
- A high temperature, red throat and feeling really ill means you probably need an antibiotic
- If you feel able to stop taking them early, you may well not have needed them
- Always take all doses for as long as instructed
- Never keep any leftovers - what's prescribed for one infection might not work for the next
- GPs can give a delayed antibiotic prescription for you to take only if things get worse
- In cases of severe illness, antibiotics can save lives
The Department of Health issued fresh guidance on antibiotic prescribing in hospitals on Friday, with doctors and nurses being urged to "think twice" before offering them to patients.
Dr Clare Gerada, chair of the Royal College of GPs, said: "Antibiotics are a wonderful thing when used properly, but they are not a cure-all for every condition, and should not be seen, or used, as such.
"The opposite is often true and, when used excessively or inappropriately, they can actually do more harm than good - reducing a patient's immunity to illnesses, or building up an immunity to antibiotics, both of which can have negative consequences for good health."
Writing in the Lancet medical journal, Prof Laura Piddock of the school of immunity and infection at the University of Birmingham warned there were global implications from the misuse of antibiotics, and drug companies' failures to develop new ones.
She warned: "The demise of antibacterial drug discovery brings the spectre of untreatable infections."