Johannesen was born in Salt Lake City, Utah, and went to New York when he was 17 to study with pianist and composer Robert Casadesus. In 1944 he made his recital debut in New York, and went on to tour with the New York Philharmonic, conducted by Dimitri Mitropoulos, and the Cleveland Orchestra, conducted by George Szell.
From 1977 to 1985 he was the president of the Cleveland Institute of Music.
Johannesen was best known for performances of works by French composers, including Gabriel Faur_, whose complete piano works Johannesen recorded. His interpretations of works by Francis Poulenc and Darius Milhaud were also admired.
One of his last projects was a piano suite called Mormoniana, which he arranged from compositions by 16 composers with roots in the Church of Jesus Christ of Latter-day Saints.