The Cowboys led 24-10 early the final quarter, but the Jets (1-0) rallied to score the game's final 17 points to come away with a season-opening victory.
Mark Sanchez (26-of-44, 335 yards, two touchdowns) found Plaxico Burress on a 26-yard TD pass with 11:56 remaining, cutting the Jets' deficit to 24-17.
The Jets tied it with 5 minutes left when they blocked a Matt McBriar punt and Isaiah Trufant ran it back 18 yards for another score.
Then with 49 seconds remaining, Darrelle Revis intercepted a Tony Romo pass and three plays later Folk, a former Cowboy, split the uprights with a go-ahead field goal, his third of the game.
Romo completed 23-of-36 for 352 yards with two touchdowns and a pick-off for Dallas (0-1), which lost for the first time in franchise history when leading by at least 14 points in the fourth quarter.