The win pulled the Twins within three games of the first-place Royals in the American League Central Division.
Rogers (10-6), who walked one and struck out seven, allowed only an RBI single to Mendy Lopez in the fifth.
After losing the final three games of a four-game series with Cleveland at home, Minnesota Manager Ron Gardenhire apologized to the Twins fans and said that someone on his team needed to step up and provide some leadership.
The Twins responded by pounding Royals rookie Jimmy Gobble (2-1), who surrendered five runs and eight hits in three innings in his first bad outing.
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