The old record for a team was set by the Oakland Athletics in 1988.
Frank Thomas' sixth home run broke a 4-4 tie in the third. It came off Tom Edens (5-3), who relieved Kevin Brown in the second, and extended Thomas' hitting streak to 10 games.
The White Sox, who won 11 of 12 games with Milwaukee, added a run off Edens in the sixth on consecutive singles by Ron Karkovice, Robin Ventura and Scott Fletcher.
"Their pitching was better," Brewers manager Tom Trebelhorn said. "They did a great job out of the bullpen. They've played better baseball than we have all season."
The Brewers took a 1-0 lead in the first when Jim Gantner led off with a triple and scored on Bill Spiers' single.
Chicago scored twice in the bottom of the first on a walk to Carlton Fisk, a single by Thomas and a two-run double by Sammy Sosa.
Milwaukee went ahead, 4-2, in the second. White Sox starter Melido Perez loaded the bases on two walks around a single by Rob Deer. Dale Sveum's double drove in the tying and lead runs and chased Perez. Gantner's single off Wayne Edwards made it 4-2.
Chicago tied the score in its half of the second. Ventura and Fletcher walked and advanced on Brown's wild pitch. One run scored as Ozzie Guillen grounded out, and Ivan Calderon's sacrifice fly tied the score.