He first appeared on the stage in 1924 in Liverpool. He acted in various stage productions until May 11th of the year that he died. He acted in the West End of London when he was young, appearing in the Terrence Rattigan play French Without Tears, which proved to be his breakthrough role.
In 1942 he divorced his first wife, Marjorie Thomas, and married actress Lilli Palmer the next year, and the two appeared in numerous plays and films. His first starring film role was as the king in Anna and the King of Siam. But the scandal of his affair with young actress Carole Landis, which led to her suicide, almost derailed his career. Harrison and Palmer divorced in 1957, and he soon remarried, to actress Kay Kendall.
Harrison was best known for his portrayal of Professor Henry Higgins in the musical My Fair Lady, based on the George Bernard Shaw play Pygmalion, especially after he reprised the role in the 1964 film version. He also starred in 1967's Doctor Dolittle.
In 1989, he was knighted.