Chicago didn't advance a runner past first base against Civale until two out in the fifth, when Zavala's single advanced Pollock to second.
The start of the game was delayed by 40 minutes because of rain.
White Sox manager Tony La Russa wasn't medically cleared to attend Tuesday's game. The White Sox are 13-7 under Cairo.Guardians: RHP Zach Plesac threw 45 pitches in a three-inning simulated game prior to Triple-A Columbus' game at Toledo with no issues. White Sox: SS Tim Anderson took batting practice and fielded grounders but wasn't sure if he would need a minor league rehab assignment before rejoining the team. "We would love for him to go, but he'll make that decision," Cairo said. Andrus took grounders at second base in anticipation of Anderson's return, which isn't likely to occur during this three-game series. "I don't have a time table on it," Anderson said. "It's just mostly how I feel.