The Patriots were soon level, however, as Brandon Bolden found a way around the defence for a simple one-yard score.
Smith's persistent interception problems surfaced again towards the end of the first quarter as his pass was picked off by Logan Ryan, who had clear space ahead of him to run for a 79-yard touchdown return.
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Smith then atoned for his earlier interception with a fine eight-yard run that took him past three defenders and just into the corner of the end zone to give his side the lead.
Folk struck a 37-yard field goal to extend the Jets' lead, but it was soon restored to three points when Stephen Gostkowski added a kick of his own at the start of the fourth quarter.
Gostkowski then slotted another kick with just 19 seconds left on the clock to force overtime for the Patriots, but it merely delayed the Jets' celebrations as Folk kicked a 42-yard field goal in the additional period to secure the win for his side.