Harrison's two-run drive down the left field line was the second of his career as a pinch hitter. It stared Pittsburgh's nine-run outburst over the seventh and eighth innings that was capped by Alvarez's sixth of the year.
McCutchen, last year's NL MVP, hit a two-run homer in the first inning in his 57th at-bat.
Gaby Sanchez also connected while Alvarez, Starling Marte, Russell Martin and Jordy Mercer each had two of the Pirates' 12 hits.
Edinson Volquez (1-0) gave up two runs in seven innings for his first win with the Pirates, scattering eight hits.