The man snatched the bag from the woman in the Coal Wynd car park in Kirkcaldy at about 15:35 on Saturday.
After her husband was knocked to the ground, and bumped his head, another woman stepped in and retrieved the bag.
The suspect was described as being white, in his 30s, about 5ft 7in tall, of scruffy appearance with frizzy hair.
He was wearing a black bomber-style jacket and grey tracksuit bottoms and was carrying a rucksack.
Det Sgt Craig Fraser, of Police Scotland, said: "This was nothing short of a cowardly attack on an elderly lady by this suspect who saw fit to struggle violently with all three people in his attempt to steal this bag.
"The last sighting of the suspect we have is of him is on the High Street near to the indoor market. Anyone who can help us identify this man is asked to call us."
Police are trying to identify a man who tried to steal an elderly woman's handbag and knocked her husband to the ground when he tried to intervene.