For most food products tested there was a "significant drop" in the levels of virus over the first 24 hours.
But in some cases traces survived for about a week, the University of Southampton team found.
"For a highly infectious agent such as Sars-Cov-2, which can be transmitted through touching contaminated surfaces and then the face, these findings are highly noteworthy," they say.
"The public may be interested in the finding that virus may persist in an infectious state, on foods and food packaging surfaces, for several days under certain common conditions."
But they add that there is no need for shoppers to take extra precautions when handling food - other than washing your hands before preparing and eating it, and rinsing fresh produce to help to remove any contamination on the surface.
They picked foods often sold loose at grocery, deli or bakery counters, such as apples, peppers, cheese, ham, olives, crusty bread and croissants.
The packaging tested included drink bottles, cartons and cans.
The amount of virus they applied was designed to simulate how much might land on food if someone who was infected coughed or sneezed near it, for example, because Covid is spread by respiratory droplets.
Breathing in infected droplets, rather than touching infected surfaces, is still the main way people catch Covid.
Anthony Wilson, microbiological risk assessment team leader at the FSA, said: "In the early stages of the pandemic, we didn't know much about how the virus would survive on different food surfaces and packaging, so the risk assessment was based on a worst-case assumption.
"This research gives us additional insight into the stability of coronavirus on the surfaces of a variety of foods and confirms that assumptions we made in the early stages of the pandemic were appropriate, and that the probability that you can catch Covid via food is very low."