Olsen was born in Ogden, Utah to Mormon parents and studied at Weber State Academy. The voice training he received there served him quite well in the years to come, both on the Broadway stage and in Hollywood. He then went to study at the University of Utah. During World War I, he sold war bonds for the United States Navy. He also studied and performed in the Eastern United States around this time. In 1923, Olsen organized the "Moroni Olsen Players" out of Ogden. They performed at both Ogden's Orpheum Theatre and at various other locations spread from Salt Lake City to Seattle.
After having worked on Broadway with shows, like The Trial of Joan of Arc, Iphegenia in Aulis, Mr. Faust, Candida, and Mary of Scotland, Olsen made his film debut in a 1935 adaptation of The Three Musketeers. His most well known role is Herbert Dunstan in 1954's Father of the Bride and its sequel, Father's Little Dividend. His other film credits include It's a Wonderful Life, Notorious, Annie Oakley, Life with Henry, Santa Fe Trail, My Favorite Spy, The Glass Key, Air Force, Madame Curie, The Song of Bernadette, Buffalo Bill, Ali Baba and the Forty Thieves, Mildred Pierce, Week-End at the Waldorf, At Sword's Point, Possessed, Life with Father, Up in Central Park, Samson and Delilah, Lone Star, Sign of the Pagan, Washington Story, and The Long, Long Trailer.
Olsen was an active member of The Church of Jesus Christ of Latter-day Saints, being a teacher of youth in the Hollywood Ward. He also was director of the Pilgrimage Play of Hollywood for several years. He died on November 22, 1954, of a heart attack and is buried in the Ogden City Cemetery.