"The message to the team was that if you don't protect the football and you don't tackle well on defense, it's hard to beat any team, let alone one of the best in the league," Rams coach Steve Spagnuolo told reporters.
The Saints (10-3) led 14-0 in the first quarter after two Brees touchdown passes to Colston.
The Rams (6-7) used two Josh Brown field goals in the second quarter to cut the deficit and were going for another score when rookie quarterback Sam Bradford's red zone pass was intercepted by Jenkins and returned 96 yards for a touchdown with 47 seconds left in the first half.
Brees completed 25 of 40 passes for 221 yards, with three touchdowns and two interceptions.
Bradford threw another interception to Jenkins in the red zone on the team's first series of the fourth quarter, and then scored the Rams' lone touchdown on their next possession on a 1-yard quarterback sneak.
It was the first rushing touchdown for the No. 1 overall pick, who completed 18 of 32 passes for 231 yards and the two costly interceptions.
"I've got to be better when we're down there and I've got to be smarter with the football," Bradford told reporters. "I just can't make those mistakes."