Conan O’Brien and his family were out to dinner in Santa Monica last year when his daughter began to screech, “Oh my God! Oh my God! Oh my God!”

“I thought a Cessna had just plowed into the sidewalk and burst into flames,” the late-night TV host recalls. “Then my son started to freak out and he was like, ‘They’re crossing the street! They’re crossing the street!’”

The source of the pandemonium was the arrival of what Mr. O’Brien’s children deemed some bigger celebrities: a few mild-mannered Mormons. The late-night TV host, who soon took a picture with them, recognized them as the stars of Studio C, a sketch comedy show out of Brigham Young University.

Studio C has achieved sizable popularity on the internet, despite—or perhaps because of—its super-scrubbed brand of clean humor, such as a skit about a soccer goalie named Scott Sterling who accidentally, and agonizingly, blocks shots with his face.

Working blue is out of the question for this comedy troupe. BYU, run by the Church of Jesus Christ of Latter-day Saints, has ranked as the nation’s most “Stone Cold Sober” place of higher learning for 20 straight years, according to the Princeton Review.