Newark are through to the second round after a stoppage-time header completed a comeback that few in the ground thought was coming.
The visitors led at half-time and hit the bar early in the second half before the home side equalised from the penalty spot on 71 minutes.
The winner came from a corner in the 90th minute, and the manager admitted afterwards: “I had already written my losing speech.”
The second-round draw takes place on Monday evening.