"It's easy to fold up and get down, but we stuck with it, fought and came up with a win," Fuld said.
Danny Santana had two RBIs for the last-place Twins, who won for just the 11th time in their last 30 games.
Perkins loaded the bases in the ninth, and it looked like Minnesota's 10-game homestand was going to end with yet another frustrating defeat.
But Perkins retired Gordon Beckham on a flyout for his 25th save in 28 chances. The Twins averted getting swept by the White Sox in a four-game series for the first time in franchise history.
"Winning one isn't a great consolation prize, but it's better than losing," Perkins said. "Hopefully it'll give us a little momentum. We came back, we battled back."
Belisario threw two pitches to Willingham, and the right-hander then was told by plate umpire Jeff Kellogg to remove a bracelet from his glove hand.
"Yeah, I got mad because I always wear it like that. Whatever," Belisario said.
Chicago manager Robin Ventura didn't think the incident had anything to do with the Twins' comeback.
"You just can't put two guys on in that situation. You put yourself in a bind," he said. "We gave them an opportunity there."
Guerra relieved and retired Eduardo Escobar on a foulout, and Santana hit a tying sacrifice fly just deep enough to left-center field. Fuld followed with a line-drive single up the middle and was thrown out at second when he tried to advance.
White Sox starter Scott Carroll allowed one run and four hits in six innings, leaving after just 78 pitches with a blister on his middle finger.
"I wasn't able to fully finish pitches like I wanted to. We thought it was best to call it after six," Carroll said.
Santana's RBI triple in the third gave Minnesota its first lead of the series. Chicago went ahead in the sixth on Alexei Ramirez's sacrifice fly and Abreu's RBI double.
When Minnesota's Oswaldo Arcia struck out ending the sixth, he snapped his bat over a knee, sending wood splinters flying around the batter's box that were picked up by a batboy.
"I would rather have him break it swinging than over his knee," Twins manager Ron Gardenhire said.
Gardenhire also hopes Sunday's win helps his team forget about a 3-7 homestand that knocked the Twins further out of contention.
"A very disappointing homestand, but to get the win today was huge," Gardenhire said. "A lot of smiles out there, a lot of relief. That was probably as intense a ninth inning as you'll get."
Twins 1B Joe Mauer (oblique) is to take batting practice at Target Field this week. ... Gardenhire wasn't worried about a setup for Paul McCartney's concert scheduled for Saturday getting in the way. "He's going to hit it right through the drums," Gardenhire joked. ... Twins SP Kyle Gibson (back) is expected to start Tuesday against Kansas City. While Phil Hughes took a line drive off a knee on Thursday he doesn't expect to miss a start. ... Ricky Nolasco (elbow) threw a bullpen session for the Twins. ... Jose Quintana (5-7, 3.15 ERA) is set to start for the White Sox on Tuesday against Detroit's Anibal Sanchez (7-4, 3.45). ... Gibson (8-8, 4.19) faces James Shields (9-5, 3.58) as the Twins open a three-game series at Kansas City on Tuesday.