Colston finished with seven catches for 105 yards, including a go-ahead, 35-yard TD pass early in the final frame and the game-clinching 28-yard catch with 7 minutes remaining.
Brees completed 36-of-46 passes for 337 yards and the NFC South leading Saints (10-3) held off a pair of late challenges to preserve their fifth straight victory.
Tennessee (7-6) lost quarterback Matt Hasselbeck to a left calf injury in the second quarter but played well under rookie backup Jake Locker, who connected on 13-of-29 passes for 282 yards and a touchdown.
Locker also provided a ground threat, carrying the ball six times for 36 yards and a score.
Scarlett Johansson steps out with fiance after pregnancy reveal
Teacher apologizes for showing sexual image of herself in class