Police said the robbery happened at Wylie’s Newsagent in Springvale Street in Saltcoats at about 15:40 on Friday.
It is understood the worker’s husband heard about the robbery and ran to the shop, where he collapsed.
He was taken to Crosshouse Hospital, but pronounced dead a short time later.
Police said the woman was uninjured but left very distressed.
A post-mortem examination to determine the cause of the 71-year-old man’s death is due to take place and a report will be submitted to the procurator fiscal’s office.
The suspect who robbed the shop is described as white, 5ft 8in tall, with a medium build. He was wearing a blue rain jacket with the hood up, blue jeans and a black scarf covering his face.
Det Insp Fraser Normansell said: “This was a very traumatic experience for the 63-year-old shop assistant who not only suffered the ordeal of the robbery itself but also witness the elderly man collapsing in the aftermath of the incident.
“It is vital that we trace the culprit and stop him from causing any further harm and distress and as such I urge anyone was in the vicinity of Springdale Street in Saltcoats around 15:40 hrs on Friday 12 January 2018 and may have seen the suspect hanging around the area to contact police at Saltcoats immediately.”