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New York's defense quickly made a play to get them off the mark though, sacking Teddy Bridgewater in the endzone for a safety, before Smith completed a couple of long passes to Percy Harvin, who was facing his former team, to give the AFC East side the lead.
Bridgewater hit his stride towards the end of the first quarter, completing a 56-yard effort to Charles Johnson. Another pass for the wide receiver was then fumbled, but Jerome Felton recovered it for a touchdown.
Nick Folk kicked a couple of field goals in the first half, including one just before the break which left the Jets just six points down heading into the third quarter.
The kicker was reliable again after the break, kicking two more field goals in the third quarter to get the Jets back on level terms heading into the fourth.
Blair Walsh, who had missed a short effort early in the game, kicked the home side back in front at the start of the final quarter.
Smith tried to find Harvin late on to steal the win for the Jets, but they had to settle for another Folk field goal to tie the scores with 28 seconds to play.
Walsh got another shot at the post, but from 56 yards out he pulled the ball well wide, forcing the game into overtime.
The Jets failed to score on their possession in the additional period, but Bridgewater made a big play to secure the win for the Vikings, tossing an 87-yard pass to Jarius Wright to clinch the victory.
As a result, the Jets slip to 2-11 this season, while the Vikings move to 6-7, leaving them third in the NFC North.