Carson: "It was so X"; Audience: "How X was it?"

Based on this paper, I think that kind of joke best falls under the category of "conversational humor." It has a strong audience-participatory element (much like a knock-knock joke does). On stage, added humor can be derived from the audience's failure or hesitation to participate in such jokes, the comic acting desperate because he's been "left hanging."

While I cannot tell you the precise origins of this specific form of audience-participatory conversational joke, I do know that it was extremely common in the Catskill region of upstate New York during the 1930s-1950s with comics like Don Rickel, Henry Youngman, Charlie Callas, and their contemporaries. That group of comics developed a lot of conventions of modern standup, and given that they often played to repeat audiences vacationing in the Catskills year after year, it wouldn’t surprise me if this form of prompting for a punchline started there.

This very much reminds me of British pantomime theatre, or at least a key trait featured in it. It incentivizes the audience to communicate with the actors, often by having them repeat (and slightly alter) statements.

Link:

Audience participation is probably the most important part of British Pantomime tradition. Even the most serious and mature members of the audience, suddenly become uninhibited in their enthusiasm to join in. The audience is actively encouraged to cheer and clap for the Hero, and to boo the Villain whenever he or she enters. Heckling is also encouraged and can lead to some hilarious ad-libs from the cast. Here’s a typical example of involving the audience:

Ugly sister: “I’m much prettier than Cinderella”
Audience: “Oh no you’re not!”
Ugly sister: “Oh yes I am!”
Audience: “Oh no you’re not!”

If anything, this back and forth is formulaic, and your description of Carson's crowd interaction is too. The actor (or in Carson's case, host) makes a statement that's clearly intended to evoke a simple and straightforward response from the audience.