It took three days to clean the pots, haul away the mountain of beer cans and sober up the last of the revelers, but by Oct. 24, 1967 everything was back to normal in the Big Bend ghost town.

The Terlingua Chili Cook-Off had been a rousing success, but no one thought at the time that they had started an annual shindig that would become a Texas tradition.

It all began that summer with the publication of an article entitled “Nobody Knows More About Chili Than I Do” in the August issue of...