Fleetwood made a storming start and were awarded a penalty on three minutes when Schumacher rounded the keeper and unleashed a goal-bound shot that was blocked by the arm of Robbie Willmott.
Willmott was shown a straight red card and Schumacher dispatched the resultant spot kick with confidence and precision.
Newport responded well to their early setback and have been the better side since falling behind. The visitors have looked particularly dangerous from set pieces.
Fleetwood could end the day on top of the League Two table if they hold on to win and results elsewhere go in their favour.