The hosts were first on the scoreboard as Nick Folk found his range early in the campaign with a 45-yard field goal.
However, Geno Smith saw a pass acrobatically intercepted by Charles Woodson later in the first quarter, allowing Derek Carr to go on and record his maiden NFL touchdown pass by finding Rod Streater, who crept into the endzone down the left touchline.
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Both sides struggled to get points on the board for a while after that, but the Jets finally did retake the lead shortly before half time as Chris Johnson reached into the endzone for a five-yard score having been found by Smith.
Folk added another field goal in the latter stages of the third quarter to give his side a six-point lead, which was extended further in the final quarter.
Chris Ivory broke through the centre of the Oakland defence before racing in for a 71-yard score, although an attempted two-point conversion was unsuccessful.
The visitors applied late pressure as they looked to mount a comeback, with James Jones leaping high in the endzone to collect a looping 30-yard pass from Carr with just over a minute remaining.
However, it wasn't enough as the Jets held out for the victory at the MetLife Stadium.