The Falcons rallied from a 27-14 deficit in the fourth quarter, going ahead on Matt Ryan's 3-yard touchdown pass to Levine Toilolo with 1:54 remaining.
But rookie Geno Smith, who threw three touchdown passes, guided the Jets (3-2) on the winning drive. He completed four straight passes and ran for an 8-yard gain before the Jets ran the clock down and sent Folk on for the last play.
Jacquizz Rodgers scored on a pair of touchdown runs, and Ryan also threw a scoring pass to Jason Snelling.
But the Falcons (1-4) are now mired in their longest losing streak since 2007.