Eaton was ruled safe on the play by first-base umpire Tony Randazzo, but Gardenhire came out to challenge the play. The call on the field stood after a review of one minute, 50 seconds.
It was the 22nd challenge issued by Gardenhire this season, and he's had 11 calls overturned.
After the review, Alexei Ramirez tied the game with a sacrifice fly before Jose Abreu scored Eaton on a two-out double to give the White Sox a 2-1 lead.
Rhett Bollinger is a reporter for MLB.com. Read his blog, Bollinger Beat, and follow him on Twitter @RhettBollinger. This story was not subject to the approval of Major League Baseball or its clubs.