Following two singles to open the inning, Cozart squared to bunt and was hit by a fastball up and in by right-hander Jimmy Nelson. He spent several moments on the ground being examined by the trainer and manager Bryan Price before going to first base.
"I escaped some big injuries getting hit on the hand batting," Cozart said. "You never know. You see guys all the time going down with broken bones. It's just a little swollen, a little numb right now. Hopefully, I will show up tomorrow and it will feel a lot better."
Brewers manager Ron Roenicke challenged the hit-by-pitch ruling, arguing that the ball had hit the bat.
"I didn't really know what happened besides that I got hit on the hand. It didn't hit the bat," Cozart said.
Ramon Santiago moved from third base to shortstop following Cozart's exit.
Cozart's status for Wednesday's series finale remains iffy.
"We won't know until tomorrow," Price said. "He'll certainly continue to get treatment tonight and tomorrow when he comes in. I don't anticipate him being able to play tomorrow, but it doesn't mean he won't be in the lineup either. We'll have to wait and see on that one."
Mark Sheldon is a reporter for MLB.com. Read his blog, Mark My Word, and follow him on Twitter @m_sheldon. This story was not subject to the approval of Major League Baseball or its clubs.