Jamey Carroll provided the game-winning sacrifice fly to give the Minnesota its fourth walk-off victory.
"It was a big win for us in the first game in this series after those guys shellacked us at their place," Gardenhire said. "It was nice to win."
Danny Valencia started the rally against right-hander Brett Myers with a single to lead off the inning. Brian Dozier then laid down a sacrifice bunt and White Sox catcher A.J. Pierzynski threw the ball away to allow the runners to advance to second and third with nobody out.
Carroll then brought home the winning run with a line drive to right field to score Alexi Casilla, who served as a pinch-runner for Valencia.
"With two strikes, I was just trying to put it in play," Carroll said. "With the infield in, you feel like you have a chance to put it past them, but fortunately I got enough on it to allow Alexi to score."
The late-inning drama came after rookie right-hander Cole De Vries struggled early, giving up four runs in the first frame. Paul Konerko came through with a two-out RBI single before Pierzynski crushed a three-run homer into the second deck in right field.
"I dug myself into a pretty good hole there," De Vries said. "It was one of those games where I didn't pitch as well as I could've obviously. It was nice the rest of the team did what they did helping me out."
The Twins came back with four runs of their own in the bottom of the first against fellow rookie Jose Quintana. Minnesota opened the inning with four straight hits that brought home two runs before Ryan Doumit reached on a throwing error from shortstop Alexei Ramirez on a potential double play that allowed a run to score. Doumit reached second on the error and scored on a two-out single from Valencia.
Minnesota tacked on two more runs in the third, when Doumit brought home a run with an RBI single before Valencia added a sacrifice fly. But Quintana settled down, as the lefty ended up going seven innings, allowing six runs (four earned) on a season-high 10 hits.
"He got in trouble early," White Sox manager Robin Ventura said. "They were hitting him. After that, he settled down and [we saw] what you're used to. They just got to him early, but not a concern that anything is wrong with him or he doesn't have the stuff."
Chicago scored in the fourth with the help of a throwing error from Dozier that allowed Ramirez to reach with one out. Ramirez ended up scoring on a single from Alejandra De Aza.
The White Sox threatened in the fifth, when Konerko was hit by a pitch to lead off the inning and Alex Rios lifted a ball deep into left-center field. But Denard Span made a leaping catch at the wall to rob Rios of extra bases and De Vries was able to get out of the inning by striking out Pierzynski and Dayan Viciedo.
"I thought it was gonna be a no-doubter, but I kept running," Span said of his spectacular catch. "I was able to time it and just leap and let my athletic ability take over."
It was the final inning for De Vries, who gave up five runs (four earned) on seven hits and a walk over five frames.
Casey Fien and Alex Burnett tossed scoreless innings in relief of De Vries, but Jared Burton gave up a run in the eighth to tie it up. De Aza brought home the run with his fourth hit to score Ramirez, who was hit by a pitch and stole second.
Left-hander Glen Perkins threw a scoreless ninth to set the stage for the walk-off win.
"Our bullpen did a super job," Gardenhire said. "Burton made one pitch that kind of hung up there and De Aza shot one to center to tie it up. But it was a big win there at the end."