After Miami had missed an opportunity to win the game with kicker Dan Carpenter drifting his 48-yard attempt wide, the Jets marched down the field with quarterback Mark Sanchez finding Santonio Holmes with a crucial 38-yard pass.
That set up Folk for the potential game-winner from 33 yards, but the Miami crowd roared in delight as the kicked was batted down.
The cheers soon turned to groans though when it transpired that Miami head coach Joe Philbin had called timeout just before Folk's kick.
That annulled the attempt and allowed Folk to retake his kick, which he comfortably converted to secure the win.
The ending was fitting for a game in which neither side seemed to have done enough to merit victory.
The Dolphins had grabbed a 10-0 advantage on a one-yard touchdown run from Daniel Thomas and a second-quarter field goal from Carpenter.
A Folk field goal before halftime reduced the deficit to seven points and that was quickly made up by the Jets on the second snap of the third quarter.
Miami had lost key running back Reggie Bush with a knee injury on the penultimate play of the half and their troubles continued after the break.
On the second play of the half, Dolphins quarterback Ryan Tannehill aimed for Anthony Fasano but his pass was intercepted by LaRon Landry who raced 18 yards into the end zone.
Then it was the Jets' turn to suffer an injury blow with their Pro Bowl corner Darrelle Revis carted off with a knee injury.
Carpenter missed a 47-yard field goal for Miami early in the fourth quarter, but the Jets got in front (20-17) for the first time with a seven-yard Sanchez pass to receiver Jeremy Kerley.
Carpenter made amends with a 41-yard field goal with 16 seconds remaining to put the game into overtime.
On a day of surprises, the San Francisco 49ers suffered a shock 24-13 loss to the Minnesota Vikings while the New Orleans Saints sunk to 0-3 with an overtime defeat by Kansas City in the National Football League on Sunday.
To add to the surprises, the Cincinnati Bengals spoiled rookie Robert Griffin III's home debut for the Washington Redskins by coming away with a 38-31 win while the Arizona Cardinals moved to 3-0 with a win over the Philadelphia Eagles.
The biggest upset though was the 49ers, who had already beaten Green Bay and Detroit and were expected to challenge for the Super Bowl.
Vikings quarterback Christian Ponder threw two touchdown passes to tight end Kyle Rudolph, including the game winner in the fourth quarter.
It was an unusually sloppy performance from the 49ers with running back Frank Gore giving up a fumble, quarterback Alex Smith throwing his first interception of the season and David Akers seeing a field goal blocked.
"We all have fingerprints on it. Now it's a matter of bouncing back and making darn sure it doesn't happen again," said 49ers head coach Jim Harbaugh.
"The Vikings did a better job than we did, all the way around. We didn't get it done. There are a lot of reasons. We're not going to start pointing fingers or making excuses."
The Saints, Super Bowl winners three seasons ago and hosts of this season's championship game, now find themselves winless after three games - a position no team has recovered from to win the Super Bowl.
The Chiefs, who had six field goals from Ryan Succop, came back from 18 points down to win 27-24 in overtime.
Their comeback began with Jamaal Charles's stunning 91-yard touchdown run and Succop's sixth successful kick sealed it.
Charles exposed the Saints' weak rush defense again by amassing 233 yards on the ground and he added 55 yards receiving for good measure.
Redskins quarterback Griffin has enjoyed plenty of plaudits this season but it was the much less heralded Andy Dalton who emerged victorious from FedEx Field after throwing for 328 yards and three touchdowns.
Cincinnati wide receiver Mohamed Sanu stole the limelight at the outset when, in the wildcat formation, he threw a 73-yard touchdown pass to A.J. Green.
The Bengals were up 24-10 at the half but a seven-yard Alfred Morris touchdown run and a three-yard pass from Griffin to Santana Moss put the Redskins right back in the game.
Dalton responded with two touchdown passes in the fourth quarter to make Griffin's two-yard touchdown run too late to change the outcome.
The league's other big-name rookie quarterback, Andrew Luck of the Indianapolis Colts, also tasted defeat at the hands of the Jacksonville Jaguars, who notched their first win of the season, 22-17.
Luck threw for 313 yards and two touchdowns with one interception for the Colts but the Jaguars won in spectacular style with quarterback Blaine Gabbert finding Cecil Shorts down the middle and he sprinted into the end zone for a game-winning 80-yard touchdown.
The Eagles had won their first two games but it was impressive Arizona who moved to 3-0, their best start since 1974, thanks to former Philadelphia quarterback Kevin Kolb providing a few points.
Kolb was effectively squeezed out by the arrival of Michael Vick but threw for 22 yards and two touchdowns while Vick fumbled twice - one of which was returned by James Sanders for a 93-yard touchdown.
The Atlanta Falcons, showing their all-round strength again, are also unbeaten after a convincing 27-3 road win at San Diego.
Falcons quarterback Matt Ryan threw for three touchdowns and 275 yards while Tony Gonzalez and Julio Jones put up 91 and 67 receiving yards each and running back Michael Turner piled on 80 yards on the ground.
Peyton Manning suffered his second straight loss with Denver as the Broncos fell to a 31-25 defeat at home to the Houston Texans.
Houston quarterback Matt Schaub threw for four touchdowns and 290 yards while Manning managed 330 yards and two scores.
Sebastian Janikowski's 43-yard field goal on the final play secured a 34-31 win for the Oakland Raiders against the Pittsburgh Steelers.
It was a tough loss for Steelers quarterback Ben Roethlisberg who finished on the losing team despite throwing for four touchdowns and 384 yards.
Joe Flacco outdueled Tom Brady in an emotional shootout and the Baltimore Ravens exacted sweet revenge against the New England Patriots with a 31-30 home victory.
The Ravens suffered heartbreak to the Patriots in last season's AFC championship game when a missed field goal attempt doomed their chances, but this time Baltimore converted the knockout blow on a 27-yard field goal by Justin Tucker as time expired.
Flacco threw for 382 yards and three scores, rallying the Ravens (2-1) from a 30-21 deficit in the fourth quarter, including a 70-yard drive in the final two minutes that set up the game-winner.
Boosting Baltimore's offense was receiver Torrey Smith, who played despite mourning the death of his younger brother in an early-morning motorcycle accident and managed to catch six passes for 127 yards and two scores.
Brady finished with 335 yards and one passing score, but could not prevent the Patriots (1-2) from losing their second straight game.
In a wild contest that featured a few controversial calls, Tucker's final kick just barely cleared the top of the right upright, causing the Patriots to protest and coach Bill Belichick to run on the field and grab the arm of an official at the game's conclusion.