Allen finished with 27 points, and Paul Pierce had 21 points and eight rebounds as Boston improved to 3-0 since Pierce returned from a right heel injury.
The Celtics got 14 points from Brandon Bass while Greg Stiemsma, filling in for Jermaine O'Neal (hamstring), had 13 points and seven rebounds.
The Wizards remained winless on the young season, falling to 0-5.
Andray Blatche led Washington with 28 points and nine rebounds while JaVale McGee added 17 points and 14 rebounds.
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