With Allen missing a second consecutive contest with a sore ankle, Boston didn't miss a beat with Bradley, who notched a career high in scoring.
Paul Pierce added 21 points and eight rebounds as the Celtics stayed hot at home with a sixth straight win in Boston.
Greg Stiesma and Kevin Garnett each added 10 points, and Rajon Rondo dished off 11 assists in the win.
The Wizards, playing a stretch of three games in three nights, fell to 0-2 during the span despite 20 points from Jordan Crawford.
John Wall supplied 12 and nine assists for Washington.