Polanco led off the fourth inning with his sixth homer of the season after going deep in both ends of a doubleheader Monday.
Rookie Yoan Moncada hustled for a double on a slow ground ball through the infield in the first inning, moved to third base on a fly ball from Jose Abreu and then scored on a wild pitch to open the scoring.
After Polanco’s home run tied it, Vargas started the fifth with a towering shot to right on an 0-2 count to go up 2-1. Rosario hit a two-run home run in the sixth inning.
Minnesota entered with a half-game lead for the second AL wild card and is attempting to hold off a pack that includes the Los Angeles Angels, Seattle, Kansas City and Texas, who all entered Tuesday within two games of the Twins.
Kyle Gibson (7-10) struck out a season-high eight batters over seven innings, allowing one run. Gibson is 6-2 against the White Sox in his career, his most wins against any opponent, and has a 2.26 ERA in those games.
Gibson stranded the bases loaded in the second and third innings, getting three straight swinging strikeouts in the third after three White Sox reached with no outs.