Morse was born May 18, 1931, in Newton, Massachusetts. He knew as a teenager he wanted to act professionally, leaving home upon graduating high school to find work in New York City.
He shot an uncredited film appearance in the 1956 drama "The Proud and Profane" before landing a role on Broadway in Thornton Wilder's "The Matchmaker" (1955-1957), which in the '60s was retooled as the iconic hit "Hello, Dolly!"
He also appeared in the Broadway shows "Say, Darling" (1958-1959) and "Take Me Along" (1959-1960) ahead of a career-making performance as J. Pierrepont Finch in the original Broadway production of the Pulitzer Prize-winning "How to Succeed in Business Without Really Trying" (1961-1965), a giddy, musical lampooning of the business world that would later be revived with Matthew Broderick (1995) and Daniel Radcliffe (2011) in the role Morse created, and for which he won his first Tony.
Other Broadway shows included "Sugar" (1972-1973), "So Long, 1974th Street" (1976), a national tour of "Sugar Babies" (1980), and "Tru" (1989-1990), for which he won a second Tony for his uncanny performance as novelist and social butterfly Truman Capote.
Morse's final appearance on Broadway came in "The Front Page" (2016-2017).
Less frequently seen in movies, he did recreate his "How to Succeed" performance in the 1967 film adaptation. He also acted in a film version of "The Matchmaker" (1958), "The Cardinal" (1963), "The Loved One" (1965), "Oh Dad, Poor Dad, Mamma's Hung You in the Closet and I'm Feelin' So Sad" (1967), "A Guide for the Married Man" (1967), "Where Were You When the Lights Went Out?" (1968), Disney's "The Boatniks" (1970), and more recently lent his voice to "Teen Titans Go! vs. Teen Titans" (2019).
Morse made many appearances on TV, both as a guest star and in early dramas. Highlights include "Alfred Hitchcock Presents" (1959 & 1960), "Night Gallery" (1971), "Love, American Style" (1971), "Fantasy Island" (1978), "Murder, She Wrote" (1985), Grandpa Munster in the 1995 TV movie "Here Come the Munsters," and "Suddenly Susan" (1998).
He was a regular on the short-lived 1968 series "That's Life," an unusual sitcom-variety show drenched in music that strongly harkened to his theater background.
Morse's most famous TV role was one that provided a cherry on top of a career that spanned seven decades, that of advertising kingpin Bertram Cooper on 58 episodes of "Mad Men" (2007-2015), casting that cleverly winked at his "How to Succeed" fame. He was Emmy-nominated five times for his work on the series.
When the show ended, his character died — but was seen performing a Broadway musical-style version of the classic song "The Best Things in Life Are Free."
Married to and divorced from dancer Carole Ann D'Andrea, with whom he had three daughters, Morse married teacher Elizabeth Roberts in 1989. They had a son and a daughter. He is survived by his wife and five children, five grandchildren, and his older brother, actor Richard Morse.