Haefliger was born in Davos, Switzerland, on 6 July 1919 and studied at the Wettinger Seminary and the Zürich Conservatory.

He soon won the attention of Ferenc Fricsay, who engaged him for the Salzburg Festival where Haefliger's world career started in 1949 with the role of Tiresias in Carl Orff's opera Antigonae.

He also sang the role of First Armed Man in Die Zauberflöte conducted by Wilhelm Furtwängler the same year at the Salzburg Festival.

In 1952, he responded to the call of Ferenc Fricsay and joined him at the Deutsche Oper Berlin where he sang the tenor parts in all Mozart operas, in Rossini's The Barber of Seville and Le comte Ory, in Pfitzner's Palestrina, the role of Hans in Smetana's The Bartered Bride, among others.

Haefliger also gave master classes in Zürich, Japan and the United States, and wrote "Die Singstimme" (Bern 1983).