With the score tied 3-3 in the bottom of the ninth, pinch-hitter Mike Baxter drew a lead-off walk off Henry Rodriguez (0-1).
Ruben Tejada then laid down a sacrifice bunt, which Rodriguez fielded and threw wildly to first, advancing the runners to second and third.
Murphy followed by lining a solid hit to right and the Mets improved to 4-0 to start a season for the first time since 2007.
Kirk Nieuwenhuis socked his first major league homer and Jon Rauch (1-0) earned the victory by hurling two innings of scoreless relief for New York.
Adam LaRoche drove in two runs for the Nationals, who fell to 2-2 on the season.