Jimenez (12-9) corralled his third victory in three starts, during which he has allowed only one earned run over 21 1/3 innings. The right-hander yielded eight hits and a walk while striking out eight Saturday.
The Tribe came into the contest trailing Tampa Bay by 1 1/2 games for the second American League wild-card spot.
White Sox starter Andre Rienzo (2-2) was roughed up for five runs on seven hits in a four-inning stint.
Josh Phegley went 3-for-3 with an RBI for Chicago, has lost five straight games.