According to one source, when Morgan's father Henry registered at junior high school, "the registrar spelled it Brasburg instead of Bratsberg.

He began acting on stage under his birth name, in 1937, joining the Group Theatre in New York City formed by Harold Clurman, Cheryl Crawford, and Lee Strasberg in 1931.

[7][8] He appeared in the original production of the Clifford Odets play Golden Boy, followed by a host of successful Broadway roles alongside such other Group members as Lee J. Cobb, Elia Kazan, John Garfield, Sanford Meisner, and Karl Malden.

In the same year, Morgan appeared in the movie Orchestra Wives as a young man pushing his way to the front of a ballroom crowd with his date to hear Glenn Miller's band play.

A few years later, still credited as Henry Morgan, he was cast in the role of pianist Chummy MacGregor in the 1954 biopic The Glenn Miller Story.

Morgan continued to play a number of significant roles on the big screen in such films as The Ox-Bow Incident (1943) with Henry Fonda, Wing and a Prayer (1944), A Bell for Adano (1945), State Fair (1945), Dragonwyck (1946) with Walter Huston, The Gangster (1947), The Big Clock (1948) with Charles Laughton, The Well (1951), High Noon (1952) with Gary Cooper and Grace Kelly, Torch Song (1953) with Joan Crawford, and several films in the 1950s for director Anthony Mann starring James Stewart, including Bend of the River (1952), Thunder Bay (1953), The Glenn Miller Story (1954), The Far Country (1955), and Strategic Air Command (1955).

(1965) with Peter Ustinov, Frankie and Johnny (1966) with Elvis Presley and Donna Douglas, The Flim-Flam Man (1967) with George C. Scott, Support Your Local Sheriff!

(1969) with James Garner, Support Your Local Gunfighter (1971) also with James Garner, Snowball Express (1972) with Keenan Wynn, The Shootist (1976) with John Wayne and Lauren Bacall, The Wild Wild West Revisited (1979) with Robert Conrad, and as Captain Gannon in the theatrical film version of Dragnet (1987) with Dan Aykroyd and Tom Hanks.

Pete and Gladys was a spin-off of December Bride (1954–1959), starring Spring Byington, a show in which Morgan had a popular recurring role.

After Pete and Gladys ended production, Morgan guest-starred in the role of Al Everett in the 1962 episode "Like My Own Brother" on Gene Kelly's ABC drama series, Going My Way, loosely based on the 1944 Bing Crosby film of the same name.

In the 1964–1965 season, Morgan co-starred as Seldom Jackson in the 26-week NBC comedy/drama Kentucky Jones, starring Dennis Weaver, formerly of Gunsmoke.

Unlike Stevenson's character Henry Blake, Potter was a career Army officer who was a firm yet good-humored, caring father figure to those under his command.

In 1986, he co-starred with Hal Linden in Blacke's Magic, a show about a magician who doubled as a detective solving unusual crimes.

In 1987, Morgan reprised his Bill Gannon character, now a captain, for a supporting role in another film version of Dragnet, a parody and homage to the original series written by and starring Dan Aykroyd and costarring Tom Hanks and Christopher Plummer.

In 1987–1988, Morgan starred in the one-season situation comedy series You Can't Take It with You as family patriarch Martin Vanderhof.

[10] Though she was left "bruised and bloodied," the case was later dismissed after Morgan attended court-ordered anger management and domestic violence counseling program.

Hunnicutt opposite Morgan in M*A*S*H, released a statement:[13] He was a wonderful man, a fabulous actor and a dear and close friend since the first day we worked together.