Kansas City (16-10-7) moved two points in front of New York for the lead in the Eastern Conference.
New York has games remaining against Chicago and Houston, while Kansas City's final regular-season contest will come against Philadelphia.
Dwyer scored from in close and Kansas City held on for its fifth victory in seven games.
D.C. United, headed for a last-place finish in the East, had its winless streak extended to 11 games.
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