The body of Sister Gertrud Tiefenbacher from the Sacred Heart Home convent in the village of Ixopo, southwest of Durban, was discovered in her room on Saturday.
Police said her hands were tied with an electric typewriter cord and she had been suffocated with a towel.
"It is also suspected that the victim was raped before she was killed," a statement said.
An undisclosed amount of foreign currency was taken from Tiefenbacher's room, as well as a vacuum cleaner, sugar and rice.
The killing has shocked the small village, which is the setting for Alan Paton's famous novel Cry the Beloved Country.
A close friend, 85-year-old Better Firmstone, told a local newspaper that Tiefenbacher taught children at a missionary school and loved her work.
"Police have not made any arrests in connection with the crime. The search is ongoing," police spokesman Thulani Zwane said.
Zwane said Tiefenbacher came to South Africa 50 years ago and had spent all her time working for the Roman Catholic Church. She was originally from Kirchberg an der Pielach in Lower Austria.
Regional police commissioner Mmamonnye Ngobeni described the murder as "barbaric", urging the community to help police arrest the suspect.
Rape is a common crime in South Africa, with 662,649 cases reported in 2013/14, according to official crime statistics.
The murder came as the South African government mobilised soldiers to keep the peace in areas hit by a spate of violent attacks targeting immigrants.