Jonas Jerebko scored 16 points to go with seven rebounds while Tayshaun Prince finished with 15 points and five assists for Detroit, which moved to 2-12 away from home this year. The Pistons won their season-high third consecutive contest and snapped a seven-game road losing streak.
Deron Williams had 34 points and seven assists for the Nets, who lost their fourth straight game.
Trailing by as many as 18 in the second half, Williams' four-point play capped a 13-4 run and made it a one-point game with 8:28 left in the fourth.
But the Pistons scored the next six points to push the lead to 84-77 with 7:15 to play. After Williams' trey made it 86-83, the Pistons scored five consecutive points to put the game away as Monroe's emphatic dunk made it 93-85 with 2:27 left.
The Nets shot just 30.4 percent in the opening quarter, but pulled down eight offensive rebounds as the game was tied at 20 after one.
With the contest knotted at 24 early in the second, the Pistons used a 9-0 spurt to take a 33-24 lead with 7:24 to play in the opening half. Walker Russell had five points while Jerebko added four during the spurt.
Brandon Knight's three-point play with a second left in the quarter gave Detroit a 51-39 lead heading into the break. New Jersey made just 1-of-10 shots from beyond the arc in the half.
A 6-0 Detroit flurry to start the third pushed the lead to 64-46, but New Jersey whittled the lead down to seven, 74-67, heading into the fourth.