Brees connected on a 40-yard effort in the first quarter to give him 48 straight games with a touchdown pass, taking him one clear of the mark set by Johnny Unitas in 1960, before he rallied his team for an overdue victory.
After establishing the new record, Brees dug New Orleans out of a 24-14 hole to help the Saints end a run of four consecutive defeats in front of a relieved home crowd that included banned head coach Sean Payton, who was allowed to attend the game.
Payton has been suspended for the entire season due to his alleged involvement in the New Orleans Saints bounty scandal.
Brees finished with 370 yards passing and four touchdown throws, including three to Marques Colston, who added 131 yards receiving for the 1-4 Saints.
San Diego quarterback Philip Rivers threw for 354 yards and two scores, but his attempt to tie the game in the final seconds stalled in New Orleans territory when he was sacked by Martez Wilson, who forced a fumble and recovered possession.
Despite the defeat, the Chargers (3-2) remain top of the AFC West division.