Brees' 40-yard pass to Devery Henderson eclipsed the mark of 47 consecutive games with a touchdown pass set by Johnny Unitas from 1956-60.
Brees' also had three scoring strikes to Marques Colston, giving him a franchise-record 52 touchdown catches with the Saints (1-4).
At Brees' request, the NFL allowed head coach Sean Payton, assistant head coach Joe Vitt and general manager Mickey Loomis — all serving various suspensions in connection with the NFL's bounty investigation — to attend the game. They and Johnny Unitas son, Joe, saw Brees pass for 370 yards.
Philip Rivers passed for 354 yards and two touchdowns for the Chargers (3-2), but had an interception and fumble in the final quarter.