The 36-year-old, who missed the Saints' previous game because of a torn rotator cuff in his right shoulder, passed to CJ Spiller in overtime to give the Saints a dramatic 26-20 victory at the Superdrome.
Dallas were first to get points on the board four minutes into the first quarter when Dan Bailey converted a 30-yard field goal, but the home side moved into the lead before the break.
Josh Hill, on the end of Brees' 399th pass, scored the opening touchdown in 11 plays, giving New Orleans a slender four-point lead at the end of the first quarter.
The visitors bounced back in the second quarter when Joseph Randle's one-yard run resulted in a touchdown, which was the only one scored by Dallas in the first half.
No touchdowns were scored in the third quarter, but Zach Hocker scored two field goals to cancel out a 38-yard kick from Bailey.
With the game tied at 13-13 at the start of the fourth quarter, Khiry Robinson swung the momentum in New Orleans' direction when he scored a one-yard running touchdown, but Dallas hit back with a touchdown of their own when quarterback Brandon Weeden found Terrance Williams with a 17-yard pass.
Overtime was required to separate the two sides, and the heroics went to Brees, who reached a milestone 400 touchdown passes when Spiller reached the endzone to give the Saints victory.
Brees completed 33 of 41 throws for 359 yards in the game.