NEW ORLEANS — For the first time in a decade, the Atlanta Falcons have an opportunity to relegate their longtime rivals in New Orleans to early-season irrelevance. The Saints have enjoyed a decisive edge in this matchup since Sean Payton and Drew Brees joined New Orleans in 2006, particularly in the Superdome, where Brees has a 7-2 record against Atlanta. Even in 2012, when the Saints were muddling through a non-playoff season overshadowed by the bounty scandal and Payton’s related suspension, New Orleans managed to drop the Falcons, who had started 8-0, to their first loss that season.