Gee (1-3) corralled his first win of the season with 5 2/3 scoreless frames, during which he allowed three hits and a trio of walks while striking out six.
LaTroy Hawkins, Brandon Lyon and Scott Rice teamed up for 2 1/3 shutout relief frames and Bobby Parnell recorded his second save with a perfect ninth for the Mets, who claimed the rubber match of a three-game set.
John Buck blasted his seventh homer of the season. Mike Baxter added an RBI in the victory.
Nationals starter Jordan Zimmerman (3-1) absorbed the loss, surrendering a pair of runs on two hits over five full innings.