The Cuban defector, who signed with the White Sox for six years and $68 million in October, connected for a two-run shot into Chicago's bullpen in right field off Brad Penny in the fifth inning.
Abreu hit 30 or more home runs in three consecutive seasons in Cuba. He also hit .360 with six home runs in 25 at-bats in the 2013 World Baseball Classic.
"It was a fastball outside and I was able to put a very good swing on it," Abreu said through an interpreter. "It was my best contact of the spring so far. ... I hit them all over the field. I hit the ball to all parts of the field. Today it was just an outside pitch and I was able to put it that way. But the important thing is I get them."
Alex Gordon led the Royals, going 3 for 3 with two runs and an RBI, while Pedro Ciriaco homered and doubled.
"I felt comfortable up there," Gordon said. "That's all you can ask for. It's great to get three hits, but it's just the mentality of being up there and seeing pitches and, at the end of the day, knowing you were ready to hit."
Royals left-hander Jason Vargas pitched three scoreless innings and has not allowed a run over five innings in his first two starts.
White Sox shortstop Alexi Ramirez, who was 0 for 8 in his first three games, went 3 for 3.
Vargas gave up five hits, but three failed to leave the infield.
"The idea is to go out there and keep them from scoring and do it as quickly as possible, so if that means getting a couple guys on and getting multiple outs at one time, I'm good with that, too," Vargas said.
Luke Hochevar went Thursday to Los Angeles for a second opinion from Dr. Neal ElAttrache on his sprained right elbow ligament. The Royals are hopeful Hochevar will not require surgery and can return to the bullpen by early June. ... Second baseman Omar Infante has inflammation in his right shoulder that bothers him when throwing.
Penny, a two-time National League All-Star who is attempting to come back after sitting out last year, gave up four runs on eight hits, including three doubles and Abreu's home run, in two innings. He has a 15.75 ERA in his first two outings.