Texas has won four in a row and increased its lead in the American League West to 3 1/2 games over the idle, second-place Los Angeles Angels.
The Rangers will open a weekend series in Seattle Friday and the Angels will be in Baltimore.
Texas starter Alexi Ogando (13-8) rebounded from recent struggles to give up just two hits and throw six shutout innings. He had gone 0-3 in his previous five starts and was taken out as a precaution after throwing 92 pitches with his team holding a 6-0 lead.
Young expanded a 1-0 Texas lead by coming through with a three-run double in the fifth and Beltre followed with his 25th homer of the season. Josh Hamilton added a solo shot in the seventh.
All of Cleveland's runs came during the final three innings. Lou Marson doubled home two runs in the ninth against Neftali Feliz and the Indians had the potential tying run in the on-deck circle before Feliz struck out both Jason Donald and Jason Kipnis to end the game.
Fausto Carmona (6-15) surrendered six runs on seven hits in six innings to suffer the loss.
Swim Week Miami Beach 2014 [PHOTOS]