The 2-0 loss, which featured seven shutout innings by Rick Porcello, also snapped a 14-game home run streak for the Twins, who lead the majors with 137 homers.
Things don't look like they'll get any easier Tuesday night for Rocco Baldelli's club when they face Red Sox left-hander David Price (4-2, 3.52 ERA).
The 2012 American League Cy Young Award winner has compiled a glossy 10-4 record and 2.62 ERA in 18 career games against Minnesota, 17 of which were starts. Price is 4-4 with a 3.08 ERA in nine career starts at Target Field.
Minnesota, which is 4-0 this season after losing back-to-back games, will counter with right-hander Michael Pineda (4-3, 5.04).
Pineda is 5-5 with a 4.23 ERA in 12 career starts against Boston.
This season, Pineda has allowed three runs or fewer in six straight starts and comes in off a no-decision in a 10-5 victory over Seattle, when he allowed two hits and one run over 5 2/3 innings while striking out four.
Boston, which sputtered to a 6-13 start to the season after winning the 2018 World Series, brings a season-high six-game winning streak into the contest.
The Red Sox, who sit in third place in the American League's East Division, trail division-leading New York by 5 1/2 games and Tampa Bay by four games.
"Just playing good baseball," Red Sox manager Alex Cora said. "We still have to get better. Just keep getting better. That's the most important thing. It's been a good six days, and there's (been) some strides, but we're not where we want to be yet."
"We're playing good baseball," Porcello said after giving up four hits and a walk while striking out eight over seven shutout innings against the top offensive team in the majors.
"We still have a lot of baseball to play. It's hard to sit here and kind of figure out where we're at. We've got to keep going. I think there's a lot of things we can get better at all the way around, and we need to continue to close the gap in the division and keep pushing forward," Porcello said.
Boston, which is six games over .500 for the first time in 2019, improved to just 17-23 against teams with winning records. By contrast, the Red Sox are 23-11 against clubs with losing records.
"We know what we have in the clubhouse," Ryan Brasier said after earning his seventh save with a 1-2-3 ninth on Monday. "Early in the season, everybody struggled a little bit. We have a good streak going right now, and just keep it rolling and do what we know that we can do."
Baldelli wasn't ready to push the panic button after his team, which finished with just five hits, was shut out for just the second time this season and for the first time since getting blanked by Gerrit Cole and the Astros, 11-0, on April 30 in Houston.
Porcello "has good stuff," Baldelli said. "I've seen him pitch many, many times and have efforts kind of like we saw. ... He hasn't had the success that he's had in his career by mistake. He's an exceptional pitcher and he was in his highest form tonight."
--Field Level Media