Brees went 34-of-41 for 392 passing yards as the Saints (7-2) rolled up 625 yards of total offense in bouncing back from last week's loss to the New York Jets.
Marques Colston returned from a one-game injury absence to catch seven passes for 107 yards with a touchdown for New Orleans.
The Cowboys (5-5), who surrendered the most yardage in team history, fell into a tie with Philadelphia atop the NFC East.
Tony Romo made good on just 10-of-24 passes for 128 yards and a score for Dallas.
Celebrity Families of 2014 [PHOTOS]