The Wizards overcame a seven-point fourth-quarter deficit and finished the game on a 15-4 run to improve to 19-16.
The Celtics, who fell to 30-5, committed 21 turnovers in the loss while Washington overcame 20 of their own and were only 2-for-18 on three-pointers.
"It is a hard game to explain and describe, other than to say it was very, very physical," Washington coach Eddie Jordan said.
"They are a very good team. They are an extremely good defensive team. All that being said, we forced 21 turnovers. We had only 13 assists but we out-rebounded them. We forced them to shoot 41 percent (from the field)."
The Wizards out-rebounded Boston 49-30 and went 27-for-30 from the foul line in their biggest win since star Gilbert Arenas was lost with a knee injury.
"I think guys realise now that just because Gilbert Arenas isn't here, the season's not over," said center Brendan Haywood.
DeShawn Stevenson scored 19 points, Caron Butler had 16 and 10 rebounds and Antawn Jamison overcame a 4-for-15 shooting night with 10 points and 16 rebounds.
Kevin Garnett led the Celtics with 19 points as Boston's 10-game road winning streak ended.
Richard Hamilton provided the late heroics as the Detroit Pistons pulled out a 103-100 overtime road win over the short-handed Charlotte Bobcats. Charlotte dressed only eight players because of injury.
Steve Nash scored 26 of his 35 points in the second half and Amare Stoudemire scored 31 points as the Phoenix Suns scored a 122-114 win over the Milwaukee Bucks, their 20th consecutive home win over the Bucks. Andrew Bogut had a career-high 29 points for Milwaukee.
Mehmet Okur scored 29 points and six Utah players reached double figures as the Jazz scored a 119-115 home victory over the Orlando Magic.
Manu Ginobili scored 22 points, Tony Parker had 21 and Gregg Popovich recorded his 600th regular-season victory as the San Antonio Spurs scored a 105-88 win over the Minnesota Timberwolves. Minnesota has lost 14 consecutive road games.