Björnstrand was born Knut Gunnar Johanson in Stockholm as son of actor Oscar Johanson and Ella Mauléon. After his education at the Royal Dramatic Theatre's acting school, he made several appearances in theatre, film and radio. Björnstrand's first collaboration with Ingmar Bergman was the 1941 theatre production of August Strindberg's The Ghost Sonata. His first major film role was in Hampe Faustman's Natt i hamn in 1943. After signing a contract with Svensk Filmindustri, he was offered mainly parts in comedies, including Bergman's Smiles of a Summer Night (1955). In 1957, however, he appeared in two dramatic roles for Bergman, in The Seventh Seal and Wild Strawberries. He regularly starred in other films by Bergman until 1968, when his work for the Director became infrequent. He also played major roles at the Stockholm Stadsteater and in private theaters. In later years, due to the consequences of a stroke, Björnstrand avoided long-term contracts, and focused on theatre and television work. His last film was Bergman's Fanny and Alexander (1982): since he was suffering from memory loss at the time, the production was a difficult one for him. Björnstrand died in Stockholm in 1986.

In 1983 he won the Ingmar Bergman Award at the 19th Guldbagge Awards ceremony.