Cookson was born in Milwaukie, Oregon, and attended the Pasadena Playhouse on a scholarship.
He appeared in the play The Heiress on Broadway in 1947, where he met his wife to-be, Beatrice Straight. He was also a producer and produced the play The Innocents on Broadway in 1950, starring his wife.  Cookson's most famous stage role was of the love struck judge in Cole Porter's 1953 musical Can Can in which he introduced the song "It's All Right With Me." "In interviews at the time, he said he was astonished at being given the part, as he had not sung for an audience since high school."
Cookson starred in several feature films during the 1940s, including G. I. Honeymoon (1945) and Fear, before moving exclusively to television during the following decade.
^Garfield, David (1980). "Birth of The Actors Studio: 1947-1950". A Player's Place: The Story of The Actors Studio. New York: MacMillan Publishing Co., Inc. p. 52. ISBN0-02-542650-8. Lewis' class included Herbert Berghof, Marlon Brando... Beatrice Straight, Eli Wallach, and David Wayne... Also Henry Barnard, Jay Barney, John Becher, Philip Bourneuf, Joan Chandler, Peter Cookson, Stephen Elliott, Robert Emhardt, Joy Geffen, William Hansen, Will Hare, Jane Hoffman, George Keane, Don Keefer, George Matthews, Peggy Meredith, Ty Perry, Margaret Phillips, David Pressman, William Prince, Elliot Reid, Frances Reid, Kurt Richards, Elizabeth Ross, Thelma Schnee, Joshua Shelley, Fed Stewart, John Straub, Michael Strong, John Sylvester, Julie Warren, Mary Welch, Lois Wheeler, and William Woodson.