However, the New York Jets would quickly improve in defense, and two field goals in the opening quarter, followed by a third successful kick from Nick Folk handed them a 9-7 advantage early in the second period.
Brady soon found another gear to guide the Patriots back into the lead with a short pass to Vereen, who took the opportunity to claim his second score of the night.
Folk responded with his fourth field goal, before New England kicker Stephen Gostkowski made his first contribution to hand the hosts a 17-12 lead at the half-time interval.
Jets quarterback Geno Smith struggled to drive his side forward in the first half, but led the visitors on an 11-play drive in the third quarter which resulted in Chris Ivory forcing his way into the endzone from a yard out.
A stubborn New York defense frustrated Brady and the Patriots for an extended spell, but the home side managed to regain the lead with another Gostkowski field goal.
The Patriots scored a crucial third touchdown with just under eight minutes remaining as Brady picked out the run of Danny Amendola in the right corner of the field.
New York refused to accept the loss without a late fight, and were back in the game when Smith found Jeff Cumberland with a 10-yard touchdown pass, but their attempt to level the scores with a two-point conversion was unsuccessful.
The Jets would get the ball for a final time in the last minute of the contest, but Folk saw his 58-yard field goal blocked by Chris Jones with just a few seconds remaining as the Patriots held on.