The cause of the accident, which occurred below the Mannlkarscharte pass near Zell-am-See, about 80 kilometres south of the central city of Salzburg, has not yet been determined.
"We think a rope may have come loose," said Anton Voithofer, head of the Red Cross rescue team.
Five helicopters were sent to the scene, and the injured climber was transported to a hospital in Schwarzach.
The victims have not yet been identified.
It was the deadliest climbing accident so far this season in the Austrian Alps, where several dozen mountaineering deaths are reported each summer.