The Jets (2-1) got the ball following a 48-yard miss by Miami kicker Dan Carpenter to begin overtime.
Mark Sanchez drove New York into field goal range with a 38-yard pass to Santonio Holmes on the second play.
Shonn Greene carried the ball twice to set up Folk's first attempt, which was blocked. But the referees ruled the Dolphins had called time-out before the play, giving the Jets kicker another chance.
This time Folk split the uprights, giving the Jets a win.
Ryan Tannehill completed 16-of-36 for 196 yards with an interception, while Daniel Thomas rushed 19 times for 69 yards and a score for the Dolphins (1-2).