With the game tied 2-2 in the sixth, Erik Kratz drew a two-out walk from Jared Hughes (1-2) and pinch-hitter Ezekquiel Carrera followed with a base hit, sending Kratz to second.
That brought up Rollins, who lined an 0-1 pitch to right field, chasing home the decisive tally.
Raul Vales (0-2) earned the win in relief of Phillies starter Jonathan Pettibone, who held the Pirates to two runs and struck out six over 5 1/3 frames in his major league debut.
Jonathan Papelbon tossed a scoreless ninth for his fourth save as the Phils claimed the opener of a four-game set.
Celebrity Families of 2014 [PHOTOS]