The Falcons had stormed back from a 27-14 deficit with a pair of fourth quarter touchdowns to grab a one-point lead with 2 minutes remaining.
But Geno Smith led the Jets (3-2) on an eight-play, 56-yard drive with time winding down, making key competitions to Stephen Hill and Jeremy Kerley and scrambling eight yards for a first down, moving New York to the Atlanta 25-yard line.
That set up the game-winning boot from Folk -- his third field goal of the game on three attempts.
Smith completed 16-of-20 passes for 199 yards with three touchdowns while adding 21 rushing yards on three carries in the victory.
Jeff Cumberland led the Jets receivers with three grabs for 79 yards and a score.
Matt Ryan went 36-of-45 through the air for 319 and two TDs for the Falcons (1-3), who dropped a third straight.
Jacquizz Rodgers scored twice on 14 carries, gaining 43 yards in the loss.