Jimenez (3-2) overcame five walks and struck out six in breaking a personal two-game losing streak for the American League Central-leading Indians, who took the rubber match a three-game series.
Chris Perez racked up his 11th save to secure the victory.
Asdrubal Cabrera stroked a two-run double and Jason Kipnis socked a solo homer for the Tribe.
Texas starter Yu Darvish (4-1) sustained his first major-league loss after beginning the season with four victories in five starts.
The Japanese star gave up four runs (three earned) on six hits in a six-inning effort, during which he fanned 11 Cleveland batters.