Escobar's two-strike hit scored Eric Hosmer, who led off the inning with a single.
The Indians rallied from a six-run first-inning deficit, but lost their sixth straight, their longest losing streak since dropping seven in a row from June 19-26, 2010.
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Aaron Crow (2-1), the sixth Royals pitcher, worked a spotless 11th inning to pick up the victory.
Reliever Esmil Rogers (1-1), the fifth of six Indians pitchers, took the loss. Carlos Santana had a home run and four RBI for Cleveland.
Indians right-hander Corey Kluber, who was making his first major-league start, faced 10 batters in the first inning and gave up six runs, six hits and a walk, throwing 43 pitches.
Alex Gordon began the Royals' six-run first with his seventh career leadoff home run. Hosmer's three-run homer to center was the big blow of the inning. Lorenzo Cain and Jarrod Dyson had run-producing singles.
Kluber, however, pitched longer than Royals veteran left-hander Bruce Chen, who was yanked after 2 2/3 innings, allowing four runs, seven hits, one walk and two sacrifice flies. In seven starts since July 1, Chen is 0-3 with an 8.73 ERA, allowing 32 runs, 50 hits and 11 walks in 33 innings.
Kluber made it through 4 1/3 innings, giving up no runs and three hits after the first.
The Indians got a run in the second when Michael Brantley led off with a double and scored on Vinny Rottino's sacrifice fly.
The Indians cut the lead to 6-4 with a three-run third, which was highlighted by a two-run single by Santana. Brantley drove in the other run with a sacrifice fly.
Santana homered with Brent Lillibridge aboard in the fifth off Everett Teaford, who replaced Chen. Santana hit an 0-2 pitch out to left.
DH Travis Hafner, who is on the paternity list, is scheduled to rejoin the Indians on Saturday at Detroit. Hafner returned to Cleveland to be with his wife Amy, who gave birth to an 8-pound boy. ... RHP Jeremy Guthrie, who is 0-2 since being acquired in a July 20 trade with Colorado, will make his third start for the Royals, on Friday against Texas. ... The Royals scored 18 runs in the first three innings of the three-game series with the Indians. ... Kluber became the seventh pitcher to start for the Indians this season. That ties Seattle for the fewest starters used in the AL. Kansas City has used a league-high 12.