The outcome further tightened the cluttered American League wild card race with Cleveland moving into one of the two post-season spots and Kansas City climbing within two games of such a position.
The game had been tied at 1-1 since the second inning when the Royals staged a two-out rally in the eighth.
Lorenzo Cain and Mike Moustakas both singled with two away and David Lough walked to load the bases.
Neftali Feliz was then called on to make his sixth appearance of the year after missing almost all of the season with arm problems.
Feliz relieved Jason Frasor (4-3) and walked Escobar on four pitches to force in what turned out to be the winning run.
Greg Holland came on in the ninth inning to protect the lead and record his 44th save. The win went to Luke Hochevar (5-2), who recorded the final two outs in the top of the eighth.