According to J.P. Hoornstra of the Southern California News Group, manager Dave Roberts made the move as a precaution.
Darvish allowed eight hits, one walk and three earned runs in six innings of work, striking out two.
He left the game with the Dodgers trailing 3-2 after surrendering a solo home run to White Sox first baseman Jose Abreu in the sixth.
It marked Darvish's worst start for the Dodgers in three tries since getting acquired from the Texas Rangers prior to the non-waiver trade deadline.
In the Japanese righty's first two outings for L.A., he went 2-0, struck out 10 batters in each game and allowed eight hits and two earned runs combined.
The Dodgers won Wednesday thanks to a three-run ninth, and they continue to run away with the top record in baseball at 85-34.
Ace Clayton Kershaw remains on the disabled list with a back injury of his own; he pitched a two-inning simulated game Wednesday, though.
Outside of Kershaw and Darvish, Los Angeles boasts impressive depth in its starting rotation, with Alex Wood, Rich Hill, Kenta Maeda and Hyun-Jin Ryu all capable of spinning gems.