Swiss said in a statement that “out of respect for the loved ones, we will not provide detailed information about our employee or the cause of death”.
After the plane made an emergency landing, one cabin crew member was flown by helicopter to a hospital in Graz and taken into intensive care.
Another cabin crew member was also taken to hospital.
All passengers were evacuated and 12 received medical attention. Swiss on Tuesday said that all passengers who were admitted to hospital had since been able to leave.
Chief operating officer Oliver Buchhofer thanked the local emergency services in Graz who had helped passengers and crew.
“This is the saddest of days for us all,” he said.
“Losing our colleague and fellow member of our Swiss team leaves me distraught and dismayed.”
He said Swiss would work “with the relevant authorities, to determine the causes involved”.
Swiss said the focus of its investigation was on the mechanical parts of the aircraft, such as the engine, but also on the use of protective breathing equipment for the cabin crew.