Schumacher broke the deadlock from the penalty spot on four minutes after Robbie Willmott had been shown a straight red card for handling his goal-bound shot.
Newport responded well to their early setback and Fleetwood were frustrated for the remainder of the first half.
However, the hosts were quickly on the front foot after the interval. It took them just three minutes to double their advantage, as Schumacher found the net again.
Schumacher completed his hat-trick with another penalty just before the hour mark.
A third spot kick of the game then allowed Chris Zebroski to net a consolation goal for the visitors.
However, Fleetwood's three-goal cushion was restored 12 minutes from time as Ball tucked home from close range following a dazzling run by Antoni Sarcevic.