Cozart was attempting to sacrifice bunt in the top of the seventh inning of a 3-2 Reds loss when Jimmy Nelson's fastball came up and in and nailed his index and middle fingers. He was able to finish the top of the inning before being replaced defensively in the bottom of the seventh.
Ramon Santiago started at shortstop on Wednesday for the road trip finale. Cozart's pending return is listed as day to day.
"I don't think hitting will be the problem. Throwing will be the problem," Cozart said.
It was the puffier index finger that hurt Cozart the most.
"It's still purple, he said. "Gripping the baseball is going to be interesting."
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.