In the Brewers' six games this season, no starter has allowed more than three runs.
"We see the guy in front of us do a great job and we want to do even better," Gallardo said after the Brewers beat the Red Sox 4-0 on Sunday, his second straight scoreless start.
Milwaukee swept the three-game series against the Red Sox, who were swept just once last year when they won the World Series. That was at Texas from May 3-5.
Sweeping the Red Sox in front of their passionate fans at Fenway Park is even tougher.
"Any time you come in here and win a series, whether its three games or just two games, it's a great series," Brewers manager Ron Roenicke said.
The Red Sox, who never had a losing record last year, fell to 2-4 and matched their longest skid of 2013.
Gallardo (2-0) pitched six innings in Milwaukee's 2-0 win over the Atlanta Braves in the season opener. He and starters Kyle Lohse, Matt Garza, Marco Estrada and Wily Peralta have a 1.65 ERA, the bullpen has a 1.02 ERA and 12 2-3 consecutive shutout innings, and the team ERA is 1.45 with just 38 hits allowed in 56 innings.
"It's been outstanding," Roenicke said. "Hopefully, this is a sign that it's going to be a great year on the mound and I think it is going to be."
The Red Sox had nine hits but stranded nine runners as they lost their first three home games for the first time since 1984.
"When we did get people on, Gallardo was able to get the ground ball," Boston manager John Farrell said.
Of the 20 batters Gallardo retired, 11 were on ground balls.
Gallardo allowed seven hits, walked none and struck out three in 6 2-3 innings in the interleague game.
"We don't get to come here very often," he said. "The fans cheer their team on nonstop and you've got to calm yourself down."
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.
"I know our hitters are up there grinding out at-bats," Lester said, "The effort's there. Obviously, they don't want to not get any runs on the board."
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 Brewers made it 4-0 in the eighth on an RBI single by Aramis Ramirez off Edward Mujica.
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. Tyler Thornburg pitched the last two innings.
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 five at-bats after going 6 for 10 in the first two games.
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