Alvarez first went deep in the fourth inning, erasing a 2-0 Pirates deficit, then struck again to cap a six-run fifth inning that broke the game open, aided by Indians shortstop Asdrubal Cabrera's fielding error.
Alex Presley added a solo shot for the Bucs, who took the rubber match of a three-game series at Progressive Field.
Tony Watson (4-0) earned the victory with 1 2/3 scoreless innings in relief of starter Brad Lincoln, who had surrendered four runs on eight hits over 3 1/3 frames.
Indians starter Jeanmar Gomez (4-6) was tagged for eight runs (four earned) on seven hits in a 4 1/3-inning stint.
Jason Kipnis went 3-for-5 with a solo homer and Shin-Soo Choo drove in a pair for the Tribe.