Moore then came to bat for pitcher Tom Gorzelanny (4-2) and bashed the first pitch he saw into the left-field stands, putting the Nats ahead to stay.
Bryce Harper went 4-for-5 with an RBI and Suzuki contributed a run-scoring single for the National League East-leading Nats, who clinched their three-game series and won for the seventh straight time in New York.
Christian Garcia, Michael Gonzalez and Drew Storen combined for two scoreless relief innings and Tyler Clippard got his 31st save despite yielding a run in the ninth.
Dickey ended up surrendering three runs on eight hits over seven frames.
Scott Hairston homered for the Mets.