The home side started the game well as they moved methodically down the field and into the redzone before quarterback Tony Romo picked out Williams on a crossing route for a six-yard score.
Graham's miss allowed the Cowboys to score a field goal when Dan Bailey cracked his kick from 51 yards through the middle of the posts.
The Cowboys went further in the ascendency when DeMarco Murray found space over the right side of the line and he used a burst of pace to find the endzone for a 15-yard score.
Just before the break, Dallas extended their lead further when Romo hit Williams on a back-shoulder fade and the wideout carried Brian Dixon into the endzone for the score as the cornerback attempted to tackle him.
In the second half, the Saints finally got themselves on the board when Graham fired a kick between the posts from 30 yards. However, the home side pulled further away when Murray notched his second touchdown of the game as he broke through the line for a 30-yard score.
Brees kept plugging away for his side and managed to get their first touchdown of the night when he faked a handoff to Khiry Robinson and then hit his tight end Josh Hill in the flat, and he ran into the endzone for the score.
New Orleans found the endzone again minutes later when Brees drove them downfield again and this time he connected with Jimmy Graham, who scrambled down the middle for a 13-yard touchdown.
Romo denied any hopes of a comeback as he and Murray put the Cowboys into the redzone and the quarterback fired to Dez Bryant on a back-shoulder fade for an 18-yard score to earn the victory.