Nielsen studied at the Neighborhood Playhouse, but appeared only once on Broadway in the short-lived 1952 play Swallows Over Sorrentino. He later toured the country with a one-man show about lawyer Clarence Darrow.

His film and television career spanned over 60 years, and he earned Primetime Emmy Award nominations for Day by Day and Police Squad.

His many other credits include Airplane, The Naked Gun, Due South, Nuts, Shaping Up, The Golden Girls, and the 1996 television version of Harvey

He is survived by his wife Barbaree Earl and his daughters, Maura Nielsen Kaplan and Thea Nielsen Disney.