Utley had three hits and four RBI, with a three-run homer and a solo shot.
Philadelphia built a 4-0 lead but the Mets caught up with a four-run sixth inning. Feliz then put the Phillies back in front with his pinch-hit shot in the seventh.
J.C. Romero (1-0) pitched a scoreless eighth inning and picked up the win.
Carlos Beltran hit a two-run single for New York, which had won its previous five games.
Reliever Pedro Feliciano (0-1), who gave up the Feliz homer, took the loss.