But the Tribe began a dramatic rally in their final at-bat when Michael Brantley singled with one out.
After Mike Aviles fanned for the second out, Giambi came up as a pinch-hitter and hammered a pitch from Addison Reed (5-4) deep into the right-field stands, handing playoff-contending Cleveland a crucial comeback victory -- their fifth straight.
Brantley also homered and Bryan Shaw (6-3) earned the decision for the Indians, who came into the contest with a one-game lead over Texas for the second and final American League wild card spot.
De Aza finished 2-for-5 with two RBI for the White Sox, who dropped a 13th straight to Cleveland.
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