Alvarez bashed a three-run homer in the first inning to set the tone and added a sacrifice fly while Garrett Jones went 2-for-4 with a two-run clout for the Pirates, who completed a three-game sweep at PNC Park.
Andrew McCutchen added a solo shot and Gorkys Hernandez drove in a pair of runs in the victory.
Eric Bedard (3-5) allowed only two hits over six innings, overcoming four walks while striking out three to break a personal three-game winless streak.
Cubs starter Matt Garza (2-3) was touched for six runs (five earned) on seven hits in a five-inning stint.
Starlin Castro belted a two-run homer for Chicago, which is mired in its worst skid since 1997.