Stafford passed for 328 yards and tossed TDs to Puka Nacua, who had 164 receiving yards, and Demarcus Robinson as the Rams (8-7), who entered the game in possession of the final wild-card berth in the NFC, won for the fifth time in six games.
Derek Carr passed for 319 yards for the Saints, including 123 to Chris Olave, and three touchdowns, but two of them came after the Saints (7-8) fell behind 30-7.
Los Angeles started their first possession at their own five-yard line, but the 14th play of a hugely impressive opening drive ended with a two-yard touchdown pass from Stafford to Nacua on fourth-and-goal.
Lucas Havrisik extended their advantage with a 20-yard field goal, but the Rams kicker missed wide left on his next attempt from 47 yards which could have stretched LA’s lead even further.
The Saints broke through when Carr connected with Rashid Shaheed for a 45-yard touchdown in the second quarter, but New Orleans then turned the ball over at the Rams 42-yard line with 56 seconds left in the first half when Carr threw an incompletion on fourth-and-five.
Five plays later, Stafford threw a four-yard touchdown pass to Robinson that increased the Rams’ lead to 17-7 at half-time.
The home side then received the second-half kick-off and drove to the Saints’ three-yard line before stalling, Havrisik firing through a further 22-yard field goal that gave them a 20-7 lead.
Kyren Williams then ran in for a 10-yard touchdown later in the third quarter and a 104-yard day for the star running back as the Rams stretched ahead further.
Havrisik added a 32-yard field goal to make it 30-7 before Carr threw a five-yard touchdown pass to Juwan Johnson that sparked a late Saints rally.
Carr also connected with AT Perry for a 35-yard touchdown and a two-point conversion to Olave to trim the lead to 30-22 with just under four minutes left, but the Rams recovered the subsequent onside kick and ran out the clock.