The Rams lead the Saints 7-0 after starting the game 7-7.
Stafford connected with Puka Nacua for his sixth touchdown of the season.
The Rams started deep in their own territory but reached their 34 in two plays. On his first carry, Kyren Williams gained 7. Tyler Higbee took a short left-side ball from Stafford for 22 yards.
Stafford converted a third-and-6 with Nacua's 17-yard catch-and-run.
After a botched start and Cooper Kupp's 3-yard loss at New Orleans' 13, the Rams stalled. A defensive pass interference call on Williams' right-side route saved the team
The umpires disagreed with Saints coach Dennis Allen's claim that the ball was uncatchable.
Next, Los Angeles needed all four downs, but Stafford threw a 2-yard ball over the middle to Nacua in the end zone for a seven-point lead.
Stafford has 68 yards on 5-of-7 possession.