It gave Lowe his first shutout since Aug. 31, 2005, and marked the first time a pitcher threw a shutout without striking out a batter since 2002.
"As we say in this game, no one's feeling sorry for us," Twins manager Ron Gardenhire said. "We have to get it done. Sure, we're not getting a lot of breaks and stuff like that, but if you start playing better baseball, breaks start going your way. Right now we're not getting it done as a group. Pitching, hitting, the whole package. We're missing too many opportunities."
Lowe did allow six hits and walked four batters, but he was helped out by his defense, which turned four double plays. Of the 29 balls put into play by the Twins, 20 were ground balls.
The Twins' lone threat came when they loaded the bases with two outs in the seventh, but Alexi Casilla flied out to deep right field to end the inning.
"I think we threw seven or eight breaking balls the whole game," said Lowe, who threw his sinker 115 times in 127 pitches, according to MLB.com's Pitch F/X data. "It's not like you can do that every game. In this park, our goal was to try to get them to hit it the other way. We threw a lot of balls down and in."
Only rookie shortstop Brian Dozier had a good day at the plate -- he went 3-for-4 with a double -- but even he was impressed by what he saw from the 38-year-old right-hander.
"He probably has the most sink I've ever seen," said Dozier, who was celebrating his 25th birthday. "He throws a good sinker. It's not too sharp, but at the same time, he misses barrels. He's being doing it forever, so that's why he's been one of the best throughout his career. He pumps the zone and works fast."
Catcher Joe Mauer also continued his struggles so far this month, as he went 0-for-3 with a walk. He's hitting just .174 (8-for-46) in May, but said he's healthy despite taking a foul tip off his left knee on April 29.
"It's just one of those things," Mauer said. "I've drawn some walks and things like that. But I'm in a funk, and I'm trying to get out of it. I think it'd be pretty obvious that I'm not feeling too good at the plate."
While Lowe was on his game, Marquis struggled yet again. Marquis surrendered five runs on nine hits and two walks over five innings to see his ERA rise to 6.68 on the year.
"It's been tough," Marquis said. "Obviously, I'm not throwing the ball the way I want. It's getting embarrassing. I need to make a change and do something different to get better results."
He was hurt by the long ball in the fifth inning, as he served up three home runs. Shin-Soo Choo started it with a solo shot to lead off the frame before Asdrubal Cabrera and Carlos Santana hit back-to-back blasts.
It marked the first time this season the Indians hit three homers in an inning, and also was the first time the Twins allowed three blasts in an inning since Sept. 25, 2010.
It came after the Indians got out to a lead in the second on an RBI single from Casey Kotchman to score Santana, who doubled to open the frame. Cleveland added a run in the third on an RBI single from Cabrera after Choo doubled.
With the loss, the Twins were swept in a rare two-game set at Target Field. They also finished their nine-game homestand with a record of 3-6.
"We keep talking about how we're in a hole, and it's a big hole," Gardenhire said. "We have a long way to go to get back, so we need to start winning some baseball games."