The Saints (1-4) were trailing 24-14 early in the second half but then reeled off the contest's final 17 points to end their season-opening four-game losing streak.
Brees (29-of-45, 370 yards, one interception) executed two long drives, hooking up with Colston on scoring strikes of 16 and 5 yards to erase the 10-point deficit.
The game was the 48th straight in which Brees had thrown a TD pass, breaking the NFL record set in 1960 by Hall of Fame quarterback Johnny Unitas.
Colston finished with nine grabs for 131 yards. Devery Henderson hauled down eight passes for 123 yards and a score in the win.
Phillip Rivers completed 27-of-42 passes for 354 yards with two touchdowns but yielded a pick-off and lost a fumble for the Chargers (3-2).
Malcolm Floyd led San Diego's receiving corps with five catches for 108 yards and Robert Meachem made two scoring grabs.