Boston ended a three-game losing streak and won for only the second time in 14 games.
Neither side scored during the final 90 seconds of the fourth quarter and Washington moved into a tie with 14 seconds to go in the extra period when Marcin Gortat hit 1-of-2 from the foul line.
Wallace then hit his game-winning basket and Washington's John Wall failed to connect on a 3-point try at the buzzer.
Wall delivered the second triple-double of his career with 28 points, 11 rebounds and 10 assists.
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