Davis' shot to left field off Erik Johnson (0-2) on a 3-0 fastball snapped a 1-all tie. That was enough offense for Cody Anderson (1-3), who struck out a career-best nine over seven innings as the Indians snapped a three-game skid.
In the opener, Brett Lawrie broke a fifth-inning tie with a three-run homer, Todd Frazier hit his 14th of the season and the White Sox held on for a 7-6 victory.
Ramirez homered on the last pitch he saw in the opener and the first of the second game for Cleveland, which finished the day with six homers.
