Folk's winning kick came after his first try -- a 56-yard miss -- was called back due to a rare penalty whistled against New England. Officials cited defensive lineman Chris Jones for unsportsmanlike conduct after illegally pushing a teammate into a Jets blocker on the play.
The field goal capped a rally in which the Jets (4-3) erased an 11-point halftime deficit, sparked by Antonio Allen's 23-yard interception return touchdown off Tom Brady.
Geno Smith completed 17-of-33 passes for 233 yards, threw for a touchdown and ran for another in the victory.
Chris Ivory anchored New York's ground game, picking up 104 yards on 34 carries.
Brady made good on just 22-of-46 throws for 228 yards without a touchdown for the Pats (5-2), whose lead over the Jets in the AFC East shrank to one game.
Tight end Rob Gronkowski returned from an injury to amass 114 yards on eight receptions for New England.