The Nats scored four times in the sixth -- all with two outs -- to break open a 1-1 game.
Ryan Zimmerman went 3-for-5 with a run scored for National League East-leading Washington, which clinched its three-game series.
Gio Gonzalez (13-5) stymied the Mets on one unearned run over seven innings, yielding only two hits while walking two and striking out four.
Tyler Clippard picked up his 17th save with a 1-2-3 ninth.
Dickey ended up surrendering five runs (four earned) on eight hits in a six-inning effort, sustaining his first loss since April 18 -- a span of 17 starts.
Jordany Valdespin socked a pinch-hit solo homer for New York.