NHS Greater Glasgow and Clyde (GGC) said the woman was one of five patients who had tested positive for the virus on the ward at Lightburn Hospital.
Influenza B was a contributory factor in her death. The health board said two other patients with the virus were giving cause for concern.
The ward has been closed to new admissions since Monday.
The decision was made due to an increase in the number of patients testing positive for the virus.
NHS GGC said there had been no new cases of influenza B on the ward since "appropriate infection control measures" were implemented.
Huma Changez, consultant microbiologist and infection control doctor at NHS GGC, said: "We would like to express our condolences to the family of the patient who has passed away. Our thoughts are with them at this difficult time.
"There is not a risk to the wider public from this isolated outbreak."
An elderly patient has died on a ward at a community hospital in Glasgow that has been affected by influenza B.