Gee (6-7) limited the Nationals to six hits, walking three and striking out four.
Shortstop Ian Desmond picked the ball up in short left field and fired it to Zimmerman, hoping to erase Murphy (3-for-5) as he advanced to third. But his throw was wild and Murphy trotted home with the decisive run.
Kirk Nieuwenhuis, Eric Young and Byrd later added RBI for the Mets, who evened their three-game series.
Nationals starter Taylor Jordan (0-1) absorbed the loss in his major league debut, allowing three runs (one earned) on five hits in a 4 1/3-inning stint.
Kurt Suzuki went 3-for-4 with a run batted in for Washington.