Töpper, the daughter of a music teacher, began her singing studies at the Graz Conservatorium while still at high school. In 1945, she began her career at the Graz Opera in the role of Ulrica in Un ballo in maschera. The first Bayreuth Festival after World War II invited her in 1951 for Wagner's Ring Cycle. The same year, her first performance at the Bavarian State Opera followed as Octavian in Richard Strauss' Der Rosenkavalier. One year later, she became a contracted member there and was in 1957 part of the world premiere of Hindemith's opera Die Harmonie der Welt.
Her significant roles include Dorabella in Così fan tutte, Fricka in Das Rheingold, Brangäne in Tristan und Isolde, Judith in Bluebeard's Castle, the title role in Carmen. She sang in all the great opera house of the world, including London, Vienna, Milan, Brussels, Amsterdam, Rome, Zurich. High points of her career were engagements at the Salzburg Festival and at the Metropolitan Opera in New York.
In 1949, Töpper married the composer Franz Mixa (1902–1994). She was a professor for singing from 1971 until 1981 at the Munich Music College (today Hochschule für Musik und Theater München) where Elisabeth von Magnus was among her students.
- 1955 Bavarian Kammersänger
- 1962 Bavarian Order of Merit
- 1985 Order of Merit of the Federal Republic of Germany, Officer's Cross (Verdienstkreuz 1. Klasse)
- 1988 Great Cross of Honour of Styria
- 1995 Bavarian Maximilian Order for Science and Art
- 2009 Mastersinger Medal (Meistersinger-Medaille), Bavarian State Opera
- Walter Herrmann, Adrian Hollaender: Legenden und Stars der Oper. Von Gigli über Callas bis Domingo und Netrebko. Leykam, Graz 2007, ISBN 978-3-7011-7571-0.