The Saints had a perfect opening drive, going 80 yards in seven plays to take a 7-0 lead on the Cowboys.
Alvin Kamara scored on a 5-yard touchdown run with 4:10 gone in the game.
The Saints faced only one third down, and Derek Carr completed a 17-yard pass to Rashid Shaheed on third-and-two.
Carr was 3-for-3 for 65 yards on the drive. The big play came from the Dallas 44 when he hit Chris Olave over the middle for a 39-yard gain to set up Kamara’s 5-yard score.
The Cowboys now get their chance to answer.