Jose Veras (3-3) took the loss after allowing three of the first four hitters to get on base in the seventh inning, including Alex Rios, whose single to center field scored the go-ahead run.
"I'm not going to point fingers at the bullpen or anything like that," Wolf said. "They fight their tails off to get outs. For me, I just need to get better and do a better job and go from there."
All three relievers who entered the game for Milwaukee allowed a run, including Tim Dillard, who surrendered an insurance run for the White Sox on an RBI double by Gordon Beckham.
Brewers manager Ron Roenicke said he knows it was the right move to take Wolf out with a man on and one out in the sixth inning, but it's difficult not to second guess the decision after Loe failed to keep the ball low in the zone, allowing two hits to the first three batters he faced.
Wolf has a 4.65 ERA in his last 10 starts, yet hasn't recorded a win since April 30.
"That's why I feel bad about making moves when they don't work out, even though I think they're the right moves," Roenicke said. "I feel bad about it.
"[Wolf] didn't throw the ball as well as he had his last few outings. But still, he's out there battling, he gives us a chance to keep us in the game, he's got the lead, and we need to keep it for him."
The go-ahead run allowed by Veras snapped a streak of 13 appearances without allowing an earned run for the reliever.
"Veras is certainly capable, Kameron Loe is capable," Roenicke said. "Those guys, they should be guys that every time we bring them in they're making good pitches and getting people out."
The Brewers shared similar success against the White Sox relievers. After trailing, 4-3, in the top of the sixth, the Brewers jumped on the White Sox bullpen for the second straight night to capture the lead. Milwaukee notched three hits off White Sox relievers to score three runs in the inning.
Rickie Weeks was hit by a pitch for a franchise-record 100th time, ending the night for White Sox starter Dylan Axelrod after 5 1/3 innings. Will Ohman took over for Axelrod and immediately allowed an RBI single to pinch-hitter Martin Maldonado to tie the game at 4.
Nate Jones then replaced Ohman and surrendered an RBI single to Cody Ransom, a sacrifice fly to Carlos Gomez and a ground-rule double to Norichika Aoki. Ryan Braun flied out with the bases loaded to end the frame.
The Brewers left eight runners in scoring position in the sixth, seventh and eighth innings.
"Bad break on Aoki's ball that it bounces over the wall," Roenicke said. "If it doesn't bounce over the wall, we get another run. That completely changes the ballgame."
Aoki matched a franchise record with four stolen bases and finished 3-for-4. He said he can't remember stealing that many bases in a game any time in his career in Japan.
"Part of my game is being aggressive, and I had fun doing that," Aoki said through a translator.
Braun extended his hitting streak to 17 games overall and 27 games in Interleague Play with an RBI single in the third inning that tied the game at 3.
Aramis Ramirez also extended his hitting streak to seven games with an infield single in the second inning, scoring the game's first run on a double by George Kottaras that was misplayed in left field by Viciedo. The White Sox left fielder redeemed himself with two RBI singles and a two-run home run in his first three at-bats.
"It's one of those where I had a great practice," Viciedo said. "I felt it at practice and that always translates into games."
Wolf, who said he was confident throwing his breaking balls for strikes, admitted he made a mistake on the curveball Viciedo took deep to right field in fourth inning.
"I remember facing him in Spring Training," Wolf said. "I threw a fastball away and he mis-hit it. Whoever was playing right field caught it right at the fence."
Wolf said if he pitches nine scoreless innings like Zack Greinke did Friday, he won't have to worry about the winless stint.
"Sometimes the only answer is just to pitch better," he said.