Escobar finished 3-for-5 and James Loney put the Rays ahead to stay with a two-run double in the first inning as Tampa Bay breezed to a win in the rubber match of a three-game series.
Jeremy Hellickson (3-2) logged the victory despite lasting just five innings. He surrendered three runs on nine hits without issuing a walk while striking out four.
Indians starter Zach McAllister (4-5) was roughed up for five runs (four earned) on seven hits over 4 1/3 frames.
Mike Aviles drove in a pair of runs for Cleveland, which remained in second place in the American League Central, one game behind Detroit.