The Patriots (5-1), trailing by four points, seemed on their way to a sure loss when the Saints' Keenan Lewis intercepted a Tom Brady pass with 2:15 left in the contest.
But after the New England defense stymied New Orleans on the ensuing possession, the Patriots got the ball back on their own 30-yard line with 1:13 remaining and Brady marched the squad 70 yards in eight plays, capped by his scoring pass to Thompkins in the corner of the end zone.
It was the only TD pass of the game for Brady, who went 25-of-43 or 269 passing yards.
Stevan Ridley rushed 20 times for 96 yards and two scores for the Pats, who ended the Saints' unbeaten streak starting the season.
Drew Breese had given New Orleans (5-1) a 24-23 lead with 3:29 left on a 34-yard TD toss to Kenny Stills.
Brees finished 17-of-36 for 236 yards with two TDs and an interception in the loss.