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Dynamo were in the ascendancy after building a 3-1 first-leg lead back in Houston last week, putting pressure on DC to chase the game from the outset.
United controlled possession in the early stages but struggled to create clear-cut opportunities.
Branko Boskovic fired a header over the top for DC after 30 minutes and his side were punished for wasting that chance just moments later.
DC went for broke in the second half but could not find a way past a steely Dynamo defence.
United finally made the breakthrough seven minutes from time through Boskovic but it was too late to mount a dramatic comeback.
Houston become the first team to book their place in the MLS Cup Final and could face Los Angeles Galaxy in a repeat of last year's final if they overcome the Seattle Sounders later tonight.