NHS Greater Glasgow and Clyde Health Board has been ordered to apologise to the family for "significant" failings in communication and compassion.
An investigation by health watchdogs found that contact with the family about the 84-year-old's condition had been "minimal".
The elderly woman, who was not named in the report, was admitted to A&E at the Victoria Infirmary after she collapsed in a care home.
Just over three hours after she was admitted a staff nurse noticed her condition had deteriorated, suggesting a serious heart complication.
But the family were not told of this development and were asked to go to the waiting room. They were later told that she was being taken for a CT scan.
While she was waiting for the scan she had a cardiac arrest and died.
A probe by the Health Service Ombudsman con- cluded that the family should have been told, well before she was taken for the scan, that her condition had worsened.
The board has been ordered to put together a plan aimed at improving communication with relatives in A&E and to apologise to the family.
The report said: "The absence of a senior mem- ber of the emergency department team talking to Mrs A's family prior to her going for the CT scan was a significant failing in communication and compassion."
A spokeswoman for NHS Greater Glasgow and Clyde said it would send a written apology to the family and that it had "initiated a number of actions to improve communications between staff and patients and their families".