The Royals broke open a 3-3 game with the big frame, during which Chris Getz socked a two-run double and then came home on an Alex Gordon two-bagger. Escobar, who had already supplied a two-run single in the second, then chased home Gordon with his triple.
Gordon finished 3-for-4 while Jarrod Dyson chipped in with three hits for the Royals, who clinched their three-game series against the Indians.
Luke Hochevar (7-9) earned the victory, allowing three runs on seven hits while walking three and striking out six.
The Tribe's Derek Lowe (8-10) dropped his fourth consecutive start, lasting only 2 1/3 innings. He coughed up seven runs on eight hits.