Dunn also singled in the second and seventh innings and walked in the fourth to lead the way for the White Sox.
Alex Rios also doubled in the fifth, and he walked twice and scored twice in his first game back after missing five with a sore lower back.
"I saw a lot of pitches and that was my goal today, to see as many pitches as I could and see if I felt good, and I did," Rios said. "I had to run a lot today."
White Sox prospect Bryan Anderson hit a solo home run in the ninth inning.
Seven Chicago pitchers limited the Brewers to three hits. Starter Hector Santiago pitched three perfect innings with five strikeouts, and after Lindstrom surrendered the Braun homer, relievers Jesse Crain, Donnie Veal, Nate Jones, Matt Thornton and Addison Reed pitched a scoreless inning apiece.
"I really like their pitching," Brewers manager Ron Roenicke said. "I think any time you come out and pitch that way, any offense is going to look a little flat."
White Sox manager Robin Ventura was pleased that Rios and Crain -- pitching in only his fourth spring training game because of a strained right adductor -- looked good.
"Winning the game is nice, but right now it's more important they get their innings and play like you normally would see them play, without fear that thinking they are going to get hurt or pull something," Ventura said.
Brewers starter Wily Peralta took the loss after allowing four runs, two earned, on four hits and three walks in 4 1/3 innings. He threw 81 pitches, 49 strikes.
"We gave them some runs," Roenicke said, referring to the Weeks error and two near-misses for usually-slick center fielder Carlos Gomez. "I think Wily was down in the zone well, the sinker was good. He got a couple of balls elevated that they hit well. But that's a good ballclub that they have. When you make mistakes, they're going to hit it well.
"[Peralta's] last three outings, I like what he's done, and this outing today, I thing was pretty good. I think this will be a good outing for him to go into the season with."
Peralta is slated to start the Brewers' third game of the season against the Rockies.
"I've been throwing pretty good pitches, pitches down in the zone, and I think I'm ready to go," he said. "It was good being back at home getting ready for the season. I haven't pitched a night game in a while."
Up next: Backup catcher Martin Maldonado and the rest of the reserves will see extended action on Saturday in the Brewers' exhibition finale. That means fans will get a look at players like outfielder Logan Schafer and Yuniesky Betancourt in the team's final tune-up before Monday's season-opener against the Rockies.