Stacy carried the ball 27 times for the Rams (6-8), who continued an impressive run against the Saints with a fourth win in six meetings.
Clemens completed 14-of-20 passes for 158 yards with scoring strikes to Lance Kendricks and Cory Harkey.
The Saints (10-4), meanwhile, lost for the second time in three road games as Drew Brees was intercepted twice. He made good on 39-of-56 attempts for 393 yards with a score.
Marques Colston hauled in eight passes for 92 yards and a TD catch.