Escobar swung and missed at the first two pitches from Tribe closer Chris Perez and then knocked a 1-2 fastball through the left side of the infield to give the Royals the win and a sweep of the three-game series.
Aaron Crow (2-1) pitched a scoreless top of the 11th for Kansas City and was credited with the win.
Carlos Santana had four of Cleveland's six hits, including a two-run homer, and drove in four runs.
Hosmer was 2-for-5 with three RBI and two runs scored for the Royals.
Kansas City scored six runs in the first inning but Cleveland tied the score 6-6 by the fifth.