For both teams, the game was overshadowed by uncertainty off the court. Doug Collins is on his way out as Philadelphia's coach, and Pistons owner Tom Gores remained noncommittal before the game on the status of team president Joe Dumars and coach Lawrence Frank.
Detroit outscored the 76ers 33-21 in the third quarter.
A person familiar with the situation has told The Associated Press that Collins and the Sixers are trying to resolve their future relationship after the coach told them he would not return for a fourth season.
It was the 36th double-double of the season for Monroe, equaling the most by a Piston since Grant Hill had 44 in 1995-96. Hill also had 36 the following two seasons.
Detroit has won four straight, equaling its longest run of the season. The Pistons trailed 54-51 at halftime but went on a 14-5 run near the end of the third quarter to open a double-digit lead.
Monroe's spinning three-point play with 10:10 to go in the fourth made it 89-75.
The 76ers trailed by only seven late in the game, but Will Bynum's 3-pointer from the left corner made it 106-96.
Dorell Wright had 22 points, eight rebounds and six assists for the 76ers. Jrue Holiday added 14 points.
Bynum scored 22 points in just over 25 minutes for Detroit. Kyle Singler contributed 16 points and Brandon Knight had 12.
Collins didn't discuss his future with the 76ers before the game. Gores met with reporters briefly and said he was disappointed with Detroit's season, but it's not clear what will happen with Dumars and Frank.
When the game ended, Frank and Collins met near midcourt for a handshake and a few words.
NOTES: Detroit rookie Andre Drummond, a game-time decision after spraining his right ankle Friday, had five points and eight rebounds before fouling out.
AP Sports Writer Dan Gelston in Philadelphia contributed to this report.