Fitzpatrick completed 11-of-22 throws and hit Lee Evans with a 2-yard touchdown pass with 13:52 left in the fourth quarter.
Bird's interceptions helped keep Carolina out of the end zone.
Evans caught five passes for 75 yards while Rian Lindell kicked two field goals for the Bills (3-4), who have won their last two contests.
Carolina's DeAngelo Williams rushed for 89 yards on 16 carries, scoring on a 15-yard fourth-quarter run. He added five receptions for 50 yards.
Steve Smith caught six passes for 99 yards while Jake Delhomme went 27-for-44 passing for 325 yards but three interceptions for the Panthers (2-4), who had won their previous two contests.