Andre Rienzo limited the Royals to two runs and five hits over six innings. He struck out a career-high eight.
The Royals’ record dropped to 5-15 against American League Central opponents.
Gordon Beckham and Conor Gillaspie, who each had three hits, singled before Dunn homered in the eighth off right-hander Aaron Crow, who had allowed two three-run homers in his past two appearances.
Beckham has hit safely in 13 of his past 15 games, while Gillaspie increased his average to .347.
Royals rookie right-hander Yordano Ventura took the loss, allowing four runs and seven hits, including a Tyler Flowers home run, in six innings.
The Royals will look to avoid the sweep against the White Sox on Wednesday night at 7:10pm at Kauffman Stadium. Pregame starts at 6:30 on NewsRadio 1350 KMAN from the Royals Radio Network.