Rex Harrison was up against some pretty stiff competition for the Best Actor Oscar® in 1965. His co-nominees included Richard Burton and Peter O'Toole for their work in Becket, Anthony Quinn for his role in Zorba the Greek, and Peter Sellers for his role in Dr. Strangelove.
Perhaps Rex Harrison's advantage came from the fact that he had played the roll of Henry Higgins on the stage for two years in New York and one year in London. And perhaps the added edge may have been that Henry Higgins reminded Rex, more than just a little, of himself.

Born on May 5, 1908 in England, Rex Harrison made his stage debut at age 16. His career was interrupted briefly by WWII, during which he was called by the Royal Air Force to serve as a flight lieutenant. When the war was over, he headed for Hollywood. His career took off in the mid 50's with his Tony Award winning portrayal of Henry Higgins in the Broadway production of My Fair Lady™. He led a very full family life with six marriages and two children. In 1989, he was knighted by Queen Elizabeth for his achievements on stage and screen.
He never retired and was still in a starring role on Broadway up to one month before he died in 1990.
Rex was actually Jack Warner's fourth choice for the role of Higgins in the film version of My Fair Lady™. It is said that when Jack Warner offered the part to Cary Grant, Grant declined and went so far as to say that he would not even go to see the film if Rex were not to play the professor.