The Cowboys rallied from an early 17-point deficit, took the lead on a 1-yard run by Tashard Choice with 5:51 to play and seemed certain of victory when they made a big play that carried deep into New Orleans territory.
That play was a 47-yard pass from Jon Kitna to Roy Williams on which Williams sped up the middle of the field in a bid for what would have been a clinching score.
Williams made it all the way to the New Orleans 11-yard line before being caught by Jenkins, who then ripped the ball out of the wide receiver's hands and claimed possession for himself.
Given that game-saving effort by Jenkins, the Saints drove 89 yards in five plays with Drew Brees hitting Lance Moore on a 12-yard pass to put New Orleans in front.
New Orleans wrapped up the win when David Buehler missed a 59-yard field goal try with 31 seconds to go.
Chris Ivory scored touchdowns on runs of 3 and 6 yards in the first quarter as the Saints took a 17-0 lead. The Cowboys trailed by 14 at halftime and then continued their comeback with touchdown runs of 60 yards on a receiver reverse by Miles Austin and 1 yard on a fourth-down play by Marion Barber.
The defending Super Bowl champion Saints improved to 8-3 and are a half game behind Atlanta in the fight for the NFC South lead. Dallas had a two-game winning streak halted, lost for the first time under interim coach Jason Garrett and fell to 3-8.
Brees was 23-of-39 for 352 yards while Kitna completed 30-of-42 for 313 without a touchdown.