For decades, the New Orleans Saints were a perennial underdog in the National Football League. Opposing teams would automatically count the "W" when the Saints appeared on their schedule. Loyal fans continued to attend the games at the Louisiana Superdome, but would wear bags over their heads as an outward sign of their embarrassment over the poor performance of their team.

Things started to change for the Saints several years ago when Coach Sean Payton and quarterback Drew Brees took over a...