Time after time, the right-hander made the big pitch to get out of a jam.
Danny Salazar allowed only two hits through six innings but was hurt by both, and the Indians unraveled in the seventh during an 8-1 defeat Tuesday night.
Cleveland bounced into four double plays, including one by Michael Bourn. It was the first time the speedy outfielder had grounded into a double play in 375 at-bats over 107 games dating to last year.
"That hurts a lot," Bourn said of the double plays. "We're going to have to be a little more selective or more aggressive on the right pitches instead of going out of the zone."
Garza (5-10) pitched six shutout innings in his return from the disabled list, and Jean Segura hit a two-run homer for his first extra-base hit in more than six weeks. Gerardo Parra had a pair of triples for Milwaukee.
Salazar (8-5) dropped his first start since the All-Star break.
"After those first two innings, and today in general, I was feeling pretty good," said Salazar, who walked three and struck out eight. "After those two innings I was able to keep the ball down and throw my secondary pitches."
Parra hit Salazar's first pitch for the first of his two triples. Parra scored on a groundout, and Segura's home run in the second made it 3-0.
Garza scattered six hits, walked two and struck out four. He held the Indians hitless in six at-bats with runners in scoring position.
Cleveland grounded into three double plays against Garza, who had not pitched since July 2 due to tendinitis in his right shoulder, and another in the eighth against Neal Cotts.
The Indians had two runners on base in the second, fourth and fifth, but failed to score during Garza's 83 pitches.
Milwaukee has won four consecutive games and 17 of 23 but remains in last place in the NL Central.
Cleveland has lost four of six.
The Indians scored on doubles by Yan Gomes and Giovanny Urshela off reliever Michael Blazek in the seventh. The Brewers broke the game open with four runs, three unearned, off four relievers in the bottom half to make it 7-1.
Parra added an RBI triple in the eighth.
"MAJOR LEAGUE" NIGHT
The Brewers commemorated the 1989 movie that filmed many of its baseball scenes at Milwaukee's County Stadium, and it drew the attention of Indians manager Terry Francona, who appeared in the movie briefly. "It was quick, so I don't think they are sending me any royalties," Francona said. The team gave away 4,000 bobbleheads of Harry Doyle, the radio play-by-play character in the movie played by Brewers broadcaster Bob Uecker.
Indians: Nick Swisher went 0 for 1 with two walks and a run while playing five innings in right field during a rehab assignment for Class A Lake County against Fort Wayne. Swisher has been on the disabled list with inflammation in his left knee.
Brewers: RHP Wily Peralta allowed one run in five innings for Double-A Biloxi in a rehab start against Mobile. On the disabled list with a strained left oblique, Peralta yielded three hits and one walk while striking out five. Before the game, Milwaukee manager Craig Counsell said no decision had been made about whether Peralta would rejoin the Brewers following his minor league start.
Indians: Rookie RHP Cody Anderson makes his fifth major league start Wednesday against the Brewers. Anderson has pitched at least 6 2-3 innings in each of his first four starts and is 2-1 with a 0.89 ERA.
Brewers: RHP Kyle Lohse, having one of his worst seasons with a 5-10 record and 6.17 ERA, will make his first start since the All-Star break. The 36-year-old Lohse has struggled against the Indians, going 8-9 with a 5.45 ERA in 24 games vs. Cleveland.