Brees made good on 23-of-33 throws with two touchdowns, passing Warren Moon (49,325) on the career yardage list. He ended the contest with 49,564 career yards.
His favorite target was Jimmy Graham, who hauled in five passes for 100 yards with a score as the NFC South-leading Saints (9-2) notched a third straight win.
Matthew Ryan threw for 297 yards on 30-of-39 efficiency for the Falcons (2-9), who occupy last place in the division.
Steven Jackson carried the ball 16 times for 63 yards and scored Atlanta's only touchdown.