It was the first career catch for Dobson, a second-round draft pick who did not play in Week 1 because of a hamstring injury.
The Jets started Geno Smith at quarterback for the second straight game while Mark Sanchez decides whether to have surgery to repair a shoulder injury sustained late in the third preseason game.
The teams traded three-and-outs, and on New York's next possession Smith found Stephen Hill deep across the middle for a 33-yard gain. But as cornerback Aqib Talib was bringing Hill down, the receiver fumbled and McCourty scooped it up and took it all the way to the Jets' 3-yard-line.
Stephen Gostkowski's field goal made it 10-0.
Smith led the Jets into field goal position, thanks largely to a 27-yard completion to Santonio Holmes that picked up another 15 yards because defensive lineman Chandler Jones hit Smith in the head as he threw. The Jets were given a 9-yard touchdown on a pass from Smith to Clyde Gates, but replays showed the ball popped out as he went to the turf.
Instead, they settled for Nick Folk's 37-yard field goal.
Both teams won their openers on last-second field goals.
In their first home game since the Boston Marathon bombings, the Patriots held a pregame ceremony to honor several of those injured by the explosions.