The World Series champion Red Sox, who never had a losing record last year, fell to 2-4 and matched their longest skid of 2013. They were swept just once last season, in a three-game series against the Texas Rangers from May 3-5.
But the Red Sox couldn't score against Gallardo (2-0), who pitched six innings in Milwaukee's 2-0 win over the Atlanta Braves in the season opener, and have lost their first three home games for the first time since 1984.
Gallardo allowed seven hits, walked none and struck out three in 6 2-3 innings as the ERA for Brewers starters dropped from 1.99 to 1.66. None of them has allowed more than three runs in a game this season.
Jon Lester (0-2) pitched well for his second straight start but received little support again. In Boston's opener, he allowed two runs on six hits in seven innings, but the Red Sox lost 2-1 at Baltimore. On Sunday, he gave up two earned runs in 7 1-3 innings.
The Brewers got all the runs they needed in the second.
Jonathan Lucroy led off with a double and took third on a bunt single by Khris Davis. Mark Reynolds then singled in front of right fielder Daniel Nava, scoring Lucroy. And when the ball got by Nava for an error, Davis scored and Reynolds took second.
Then Lester settled down, retiring 15 straight batters before Davis doubled with one out in the seventh. Reynolds walked, Lyle Overbay struck out and Jeff Bianchi singled in Davis, making it 3-0.
Davis went 2 for 4 after getting four hits in a 7-6 win in 11 innings Saturday night, when he scored the winning run after his double.
The Red Sox threatened in the seventh after Gallardo retired the first two batters. Jackie Bradley Jr. doubled and took third on an infield single by David Ross. Zach Duke replaced Gallardo and ended the threat by retiring Nava on a fly to center.
Boston manager John Farrell challenged a call for the first time this season, but replay showed umpire Tim Timmons was correct in calling Bradley out on a close play on a ground ball in the second.
Ryan Braun reached on an infield single with two outs in the first, ending a 0-for-14 slump, but was stranded. He singled again in the eighth, stole second, took third when catcher Ross' throw went into center field and scored when Ramirez singled.
NOTES: Of Lester's first eight pitches, six were strikes and two were outs. ... Boston 3B Will Middlebrooks went on the 15-day disabled list with a strained right calf. ... The Red Sox open a three-game series at home Monday night, with John Lackey (1-0) pitching against Tanner Scheppers (0-0) of the Texas Rangers. The Brewers' game scheduled in Philadelphia on Monday was postponed because of expected inclement weather. .. Milwaukee's Carlos Gomez was hitless in four at-bats after going 6 for 10 in the first two games.