The Giants scored on two touchdown passes from rookie quarterback Jaxson Dart to Theo Johnson. The Saints (0-4) earlier scored on 28- and 53-yard field goals from Grupe. He also missed from 52 yards with 1:25 to play.
Rattler was 16 of 24 for 187 yards while running back Alvin Kamara had six carries for 21 yards. Shaheed had three catches for 106 yards and the touchdown.
Taysom Hill and tight end Foster Moreau made their season debuts after recovering from serious knee injuries but didn't have a big impact in the opening half..
Saints safety Justin Reid left the game in the first quarter with a concussion.
The Giants (1-3) will get the ball to start the second half.