Gostkowski then launched a 21-yard kick between the posts the extend the home side's lead.
The Jets thought that they had got on the board when Clyde Gates threw himself forward to catch Geno Smith's pass, but a review showed that the wide receiver lost control of the ball after hitting the ground.
However, Nick Folk nailed a 37-yard kick to reduce the deficit before the end of the first quarter.
Michael Buchanan sacked Smith as the Jets looked for a route back into the game deep in the second quarter, before Gostkowski restored the Pats' 10-point advantage with his third successful kick of the evening.
The Pats had a touchdown of their own chalked off 24 seconds before half time, as replays showed that Kenbrell Thompkins caught Tom Brady's deep pass.
Gostkowski's boot let him down from 43 yards moments later, which meant that the Pats led 13-3 at the midway point.
Bilel Powell scored the Jets' first touchdown to breath fresh life into the contest with five minutes remaining in the third quarter.