Garza (1-2), who pitched part of last season with the Chicago, retired the Cubs in order in four of the first six innings. He gave up two runs and four hits while striking out seven and walking one.
Francisco Rodriguez pitched a scoreless inning to pick up his 10th save in 10 chances.
Cubs starter Carlos Villanueva (1-5) gave up five runs and 11 hits in five innings.
The Brewers when Cubs left fielder Junior Lake appeared set to make a routine catch of Scooter Gennett's line drive but the ball sailed past Lake's glove and rolled to the fence, allowing Gomez to score.
Braun's single drove in Gennett, who scored as Lake airmailed a throw to the plate, giving the Brewers a 2-0 lead.
The Cubs cut the lead to 2-1 in the third on Emilio Bonifacio's run-scoring single.
The Brewers extended their lead to two runs again in the bottom half of the inning. With Gomez on first, Braun hit a ground ball deep in the hole at short. Cubs' shortstop Starlin Castro made a strong throw and umpires initially ruled Braun out at first. After a delay of two minutes and 20 seconds, Braun was ruled safe. Jonathan Lucroy followed with a single to score Gomez.
Overbay led off the fourth with a homer, giving the Brewers a 4-1 lead. Aramis Ramirez's sacrifice fly in the fifth extended the lead to 5-1.
Ryan Sweeney's ground-out plated a run for the Cubs in the seventh.
Notes: The Brewers unveiled a statue of Hall of Fame radio announcer Bob Uecker, situated in the last row of the upper deck behind home plate, that pays tribute to the classic Miller Lite ad campaign in which he starred. .. Rodriguez tied Robb Nen for 18th place with his 314th career save. ... Brewers manager Ron Roenicke applauded Major League Baseball's attempt to eliminate the confusion that has surrounded the ''transfer'' rule this season by returning to the previous interpretation for when a fielder loses possession of a ball while trying to shift it from his glove to his hand. ''I like what they did. I think it makes sense and it is the right way,'' Roenicke said. ... Chicago reliever Neil Ramirez made his Major League debut. ... The Brewers' Marco Estrada (1-1) will face the Cubs' Travis Wood (1-2) on Saturday.