After the game, manager Terry Francona said he didn't think the injury would force Drew to miss any time.
"He got a pretty good shiner there, it was starting to affect his vision," Francona said.
The Red Sox say Drew hurt his left eye after fouling off a pitch that ended up hitting him in the face in the batting cage. He started the game and struck out looking with two runners on to end the first and made one putout in right field before being replaced by Darnell McDonald.
The Red Sox are already shorthanded in the outfield with Carl Crawford on the disabled list with a hamstring injury.
Drew is batting .233 with four homers and 18 RBIs in 61 games.
Information from The Associated Press was used in this report.