Jimenez (5-2) won for just the second time in a month, escaping from a messy third inning to secure the victory. He gave up two singles, a home run and a walk in the third, but surrendered just one run.
Cleveland took the lead on a two-run homer by Mike Aviles in the second. Jason Kipnis hit a solo shot as part of a two-run Cleveland fifth.
The Indians won the final two games of the three-game series after having lost eight in a row. Texas has lost five of seven.
Nick Tepesch (3-6) was charged with the loss, giving up five runs on eight hits in five innings.