He was admitted at the age of sixteen to the Curtis Institute of Music in Philadelphia where he studied with Gregor Piatigorsky, who engaged him as a teaching assistant in 1949.
From 1950 to 1953, Bengtsson taught his own cello class at the Institute, before being appointed to the Royal Danish Academy of Music in Copenhagen.
Bengtsson was a member of the Royal Swedish Academy of Music and was conferred the title Chevalier du Violoncelle by Indiana University in 1993.
In November 2006, Danacord released the DVD The Cello and I, which presented a comprehensive portrait of Bengtsson's career on the seventieth anniversary of his debut.
Bengtsson was the subject of a sculpture (The Musician) in 1970 by the Icelandic sculptor Ólöf Pálsdóttir, of him playing the cello.