Drew had bunted a ball off his left eye before the game while in the batting cage.
"He got a pretty good shiner there, it was starting to affect his vision," Boston manager Terry Francona said.
In his one at-bat, Drew struck out looking against Pirates pitcher James McDonald.
Darnell McDonald replaced Drew in right field in the second inning.
Francona said he didn't think Drew would miss any time because of the injury.
Laura Myers is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.