Monroe grabbed 10 rebounds while Jonas Jerebko and Tayshaun Prince each netted 13 points as the Pistons notched a third straight win.
Detroit managed only 38 percent shooting from the field but used a tough defense to score 40 points off 24 Boston turnovers.
The Celtics played without Kevin Garnett, who missed the game because of personal reasons.
Boston's Rajon Rondo was ejected late in the third quarter for flipping the ball at the referee.
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