Kupper was born at Glatz (now Kłodzko) in Lower Silesia. She studied in Breslau and was a music teacher there before making her operatic debut in 1935. She then appeared in Schwerin (1937–38), Weimar (1938–40), Hamburg (1940–46), Munich (1946-61). She sang Eva in Bayreuth, in 1944, and returned as Elsa in 1960. She created Danae in Richard Strauss's Die Liebe der Danae in Salzburg, in 1952.
Kupper was especially admired as Countess Almaviva, in addition to Wagner and Strauss roles, she also gained considerable acclaim as Aida and Desdemona. A sensitive and warm-voiced singer, she retired in 1961, and taught at the Music Conservatory in Munich. She died in Munich.
- Lohengrin - Lorenz Fehenberger, Annelies Kupper, Helena Braun, Ferdinand Frantz, Otto Von Rohr - Bavarian Radio Chorus and Orchestra, Eugen Jochum - Deutsche Grammophon (1953)
- Aida - Annelies Kupper, Max Lorenz, Margarete Klose, Rudolf Gonszar, Otto Von Rohr - Frankfurt Radio Chorus and Orchestra, Kurt Schroder - Walhall Eternity (1952) sung in German
- Le guide de l'opéra, R.Mancini & J.J.Rouveroux, (Fayard, 1986), ISBN 2-213-01563-5