Dunn's blast in the third inning doubled Chicago's lead to 4-0 and Gillaspie's clout in the seventh blunted a Royals comeback try and handed the White Sox a 6-3 advantage.
Marcus Semien and Jordan Danks each added a solo homer as Chicago ended a four-game skid.
Erik Johnson (3-2) earned the decision after allowing three runs on five hits over 5 1/3 innings, overcoming three walks while striking out four.
Addison Reed secured the triumph with his 40th save.
Kansas City's Yordano Ventura (0-1) was hung with his first major-league loss, allowing four runs on six hits in four innings.
Billy Butler and Salvador Perez each smacked a two-run homer and Mike Moustakas chipped in a bases-empty blast for the Royals.