Curtis had three receptions for 22 yards and Stephen Jackson rushed for 97 yards and two touchdowns on 20 attempts, as the Rams snapped a three-game losing streak and improved to 3-4.
New Orleans' Aaron Brooks was 18-of-39 passing for 230 yards, two touchdowns and one interception.
Former Ram Az-Zahir Hakim caught six passes for 100 yards and one touchdown, and Antowain Smith added 56 yards on 14 carries.
The Saints sank to 2-5 with their third straight setback.
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