The Rangers, who did not suffer back-to-back losses during the first month of the season, have dropped three in a row. Cleveland, which went into the evening with the only winning record in the American League Central, has won three straight.
A two-run homer by Shin-Soo Choo in the second and Jack Hannahan's solo blast in the third staked Cleveland to a 3-1 lead.
Johnny Damon added a two-run triple in the seventh.
Jeanmar Gomez (2-1) got the win with a seven-inning effort, during which he gave up three runs on eight hits. Chris Perez worked the ninth for his 10th save in 11 chances.
The Texas pitching staff led the American League in ERA through April, but during the current three-game losing streak the Rangers have allowed 25 runs.