It was the fifth straight setback for the Royals and their eighth loss in the last 10 games, the latest slide in a season of extreme stretches of wins and losses.
Do the Royals have another hot streak in their system?
"They're grown men, they've got to pull themselves back in. It's time," said a grim-visaged manager Ned Yost. "We're going to have to find a way to get out of this. And get out of it real quick."
Gillaspie led off the 12th by belting an 0-1 pitch from Luke Hochevar over the right-field wall, just out of the reach of a leaping Justin Maxwell.
"I had a pretty good bead on it. I just tried to get over the wall and make a play on it," Maxwell said.
But over it went, although the ball bounced up, ricocheted off the back wall and returned to the field, causing a moment of uncertainty for White Sox third-base coach Joe McEwing.
"I kind of stopped running at second, because I didn't see anybody going after the ball," Gillaspie said. "Then I looked at Joe and he was waving me. You never know."
The Royals opened their half of the 12th with Billy Butler's walk and pinch-runner Jarrod Dyson's steal of second base. But White Sox closer Addison Reed retired the next three batters, and what remained of the 13,363 fans trudged to the exits.
It was Chicago's sixth straight victory, and Reed had a save in each one, making him the first pitcher to save six consecutive games since the Dodgers' Eric Gagne in 2003.
This was a game that the Royals led, 3-0, after a productive fifth inning against White Sox left-hander Jose Quintana. The bases filled up on Butler's walk and singles by Maxwell and Mike Moustakas. Emilio Bonifacio continued the trend with an RBI single. Two more runs scored on successive sacrifice flies by Alcides Escobar and Jamey Carroll.
Meanwhile, Royals starter James Shields reeled off five scoreless innings.
"This one was really frustrating. Shields really had his good stuff. I think they hit two balls hard off him," Yost said. "The rest were well-placed singles to get their runs. But I'd have paid a lot of money for a well-placed single in the later innings of that game."
Shields ran into trouble in the sixth inning, and the White Sox reduced his lead to 3-2. Jordan Danks belted a double into the right-center gap. After an out, back-to-back singles by Gordon Beckham and Alexei Ramirez pushed Danks home. The second run scored on Dayan Viciedo's two-out single.
The White Sox snagged a 3-3 tie in the next inning. Gillaspie singled, took second on Shields' wild pitch and scored on Josh Phegley's double -- a ground ball that ricocheted off first base and bounded into the outfield.
"I didn't do my job right there," Shields said. "I made a great pitch to Gillespie off the end of the bat, right over Escobar's head and then the next pitch I got a [wild pitch] and then next pitch [the ball] hit the bag. It's unfortunate."
That was Shields' seventh and last inning. He gave up three runs and nine hits with no walks and eight strikeouts.
Reliever Kelvin Herrera was the immediate beneficiary of diving catch made by Gordon on Ramirez's liner toward the left-field line.
"I had to do something," Gordon said.
Royals closer Greg Holland came out of mothballs, pitching for the first time since a double-save day last Friday at Detroit, and worked a scoreless ninth. After a two-out, two-base error by third baseman Moustakas, Holland absorbed a shot to the thigh off the bat of Danks and the ball rebounded to catcher Salvador Perez. He pounced on the ball and threw out Danks on a close play at first.
Holland was fine, Yost reported.
The Royals had a nice opportunity in the 11th. Carroll walked on a 3-2 pitch from left-hander David Purcey, who followed that by hitting Gordon in the shoulder with a pitch.
Asked if he considered having Eric Hosmer try a sacrifice bunt at that point, Yost responded: "No."
After Hosmer popped to center, Purcey was relieved by right-hander Jake Petricka, making his Major League debut. He did his job, getting Perez to bounce into an inning-ending double play.
When Gillaspie popped his home run, Petricka was rewarded with his first big league victory.
Afterward, the Royals were left pondering how to get out of their latest funk. Their 2-8 swoon followed a 19-5 surge immediately after the All-Star break. They remain 9 1/2 games behind Detroit in the American League Central and fell to seven games back in the Wild Card race.
"The season's not over yet," Maxwell said. "Have a short-term memory and get back and be ready tomorrow."
Shields expressed confidence the Royals could rebound.
"We've got a lot of character in this team and we're going to keep fighting till the end," he said. "We don't have any quit in us and that's one thing that we've done all season long: We haven't quit at all. We're going to keep grinding it out. We've just got to come up with some wins right here soon."
"We better or else it's going to be a long last month," Gordon said. "So, hopefully, we can start playing a little better and maybe make a run. Who knows?"