But New York's Val Pascucci tied it with a pinch-hit, solo homer in the next half-inning and the Mets grabbed the lead in the eighth when Ruben Tejada singled with one out, stole second and scored on Wright's grounder down the third base line off Brad Lidge (0-2).
Dickey finished with a no-decision, yielding one run on three hits over seven innings, walking one and striking out four.
Bobby Parnell (4-6) was credited with the win after getting the last out in the eighth and Manny Acosta earned his third save despite a shaky ninth for the Mets.
The National League East champion Phillies, meanwhile, dropped their seventh straight.
Philadelphia starter Cole Hamels allowed one run on four hits over seven innings, yielding three walks and striking out seven.