Rodgers (31-of-41 passing, 319 yards, interception) found Jordy Nelson on an 11-yard scoring strike with 7 minutes left to put the Packers (2-2) ahead for good.
New Orleans, however, then got a break when the referees ruled the Saints' Darren Sproles didn't fumble the ensuing kickoff, determining instead that he was down by contact.
The call allowed New Orleans to retain possession and begin a do-or-die drive that reached the Packers' 25-yard line.
Hartley's subsequent 48-yard attempt was wide, and the Saints fell to 0-4 to begin a season for the first time since 2007.
New Orleans' Drew Brees went 35-of-54 for 446 passing yards and three touchdowns without an interception.