WASHINGTON — It was all because of a 1978 appearance on Johnny Carson’s “Tonight Show,” the story goes, that David Letterman found his audience.

Mr. Letterman, an unknown stand-up comedian then, appeared relaxed and confident telling jokes in a short set, then bantering in the guest chair next to Mr. Carson’s desk. Landing on the show, and earning Mr. Carson’s approval, meant he had made it.

“I have a feeling with your shot on this show tonight,” Mr. Carson said at the end of Mr. Letterman’s segment, “that you’re going to be working a lot outside of the Comedy Store,” referring to the West Hollywood club that was a haven for young comics.

One place Mr. Letterman would go — 39 years later — was the John F. Kennedy Center for the Performing Arts, where on Sunday he accepted the Mark Twain Prize for American Humor, one of comedy’s top honors. While collecting the award, Mr. Letterman referred to his timely visit on Mr. Carson’s show, which was played on a screen during the award ceremony.