The Pirates scored three times late in the game to erase a 3-2 deficit. After tying the game in the seventh on a Rafael Perez wild pitch, Pittsburgh pushed across a run in the next frame when Andrew McCutchen walked and Garrett Jones singled off Jensen Lewis (2-2), setting up the Alvarez sac fly.
The Pirates added an insurance tally later in the frame when Jason Jaramillo drew a walk and Crosby reached Indians reliever Tony Sipp for an RBI single.
Octavio Dotel hurled a scoreless ninth inning to collect his 14th save and give the Pirates the rubber match of a three-game inter-league set at PNC Park.
Starter Justin Masterson allowed two runs on five hits over six innings of work for the Indians, who dropped their sixth game in seven tries.