Sweeney homered in the first inning and DeJesus went deep in the fourth to put the Royals ahead 4-to-1. DeJesus had three hits and scored twice to send the Royals to their third win in four games against the Twins this season.
Odalis Perez evened his record at 2-and-2, giving up three runs and five hits in six innings. The Royals' bullpen followed with three innings of one-hit relief, with Joakim Soria retiring five batters for his third save.
Sidney Ponson served up the Royals' two homers and fell to 1-and-3, allowed seven hits in six innings.
Twins outfielder Torii Hunter hit a solo blast in the first to extend his hitting streak to 12 games.