Henry's contributions to film included his work as a co-writer for Mike Nichols's The Graduate (1967) for which he received a nomination for the Academy Award for Best Adapted Screenplay.

He co-created Get Smart (1965–1970) with Mel Brooks for which he received the Primetime Emmy Award for Outstanding Writing for a Comedy Series.

At 15 years old, he made his professional acting debut in a Broadway production of Life with Father, which later toured theaters in Brooklyn, Long Island, and the Bronx.

He served in West Germany first as a helicopter mechanic[6] and then transferred to Special Services, where he toured with the Seventh Army Repertory Company, performing in a play he both wrote and directed.

[5] Henry joined the improvisational comedy group the Premise, whose ranks included George Segal and Theodore J. Flicker,[6] performing in the West Village in Manhattan.

[5] From 1959 to 1962, as part of an elaborate hoax by comedian Alan Abel, he made public appearances as G. Clifford Prout, the quietly outraged president of the Society for Indecency to Naked Animals,[7] who presented his point of view on talk shows.

"Prout" received many letters of support from TV viewers, and even some unsolicited monetary donations, all of which were invariably returned, as neither Henry nor Abel (who had no intention of following through on the Society's stated aims) wanted to be accused of raising money fraudulently.

Two TV projects created by Henry had short runs: Captain Nice (1967) with William Daniels as a reluctant superhero, and Quark (1978), with Richard Benjamin in command of a garbage scow in outer space.

[1] Henry shared an Oscar nomination with Calder Willingham for their screenplay for The Graduate (1967), in which he also appeared in a supporting role as a hotel desk clerk.

[5] He co-directed Heaven Can Wait (1978),[12] the remake of Here Comes Mr. Jordan, with the movie's star Warren Beatty and appeared in the film as an officious angel, reprising the character originally played by Edward Everett Horton.

Celebrity impersonations on SNL Henry died of a heart attack at Cedars-Sinai Medical Center in Los Angeles on January 8, 2020, at age 89.