Plouffe connected against Bryan Shaw (0-1), sending a drive into the second deck in left field and getting swarmed by teammates as he hit home plate. Chris Herrmann had two hits, an RBI and some slick baserunning to help the Twins win their third straight.
Kluber was perfect through five innings, but uncharacteristically erratic after that. He gave up two runs on three hits and struck out eight in eight innings. He also threw two wild pitches and hit a batter.
It was the second time this season that Kluber took a no-hit bid into the sixth inning. He also did it on opening day at Houston.
Blaine Boyer (1-1) went 1 1/3 innings for the win.
The slumping Plouffe hit his second homer of the season. he began the day at .133 and was hitless in his first four at-bats of the game.
Mike Pelfrey, who gave up one run on three hits and walked four in five innings for the Twins. His wild pitch in the fifth allowed the Indians to get on the board and Moss' solo homer in the sixth off Tim Stauffer gave Cleveland a 2-0 lead.
Kluber was in complete control through five innings against a Twins lineup with five players hitting .200 or lower.
Then, seemingly out of nowhere, the Indians and Kluber came unglued in the sixth.
Oswaldo Arcia, who had struck out 12 times in 17 career at-bats against Kluber, broke up the perfection with a single to right field and moved to second on a wild pitch. Herrmann followed with an RBI single and moved up to second on a throwing error by Moss in right field.
Then shortstop Jose Ramirez fielded a hard grounder from Jordan Schafer and tried to get Herrmann at third base. Herrmann made a nifty slide to avoid a tag, then scored on Kluber's second wild pitch of the inning.
Kluber had only three wild pitches all of last season and would have matched that number had his errant delivery in the eighth not hit Eduardo Escobar in the left foot.
Michael Brantley returned to the Indians' lineup after missing four games with a bad back and went 2 for 5 as the designated hitter.
Indians: Carlos Carrasco threw a bullpen three days after being struck in the face by a line drive. Manager Terry Francona said everything went well and Carrasco is on track to start Monday.
Twins: Escobar left the game after being hit in the eighth. ... LHP Brian Duensing was placed on the 15-day disabled list with an intercostal strain, an injury to the muscles in his rib cage. GM Terry Ryan said Duensing first felt tightness on Monday, but tried to pitch through it. Caleb Thielbar was recalled to take his place.
Indians: RHP Danny Salazar will make his season debut in game two of the series after being called up from Triple-A Columbus. Francona said the plan is to leave him in the rotation for a while and keep Zach McAllister in the bullpen.
Twins: RHP Phil Hughes (0-2, 5.25) will look to get on track after a sketchy first two trips to the mound this season. Hughes is the team's ace, and is being counted on more than ever after Ervin Santana was suspended for 80 games and Ricky Nolasco hit the disabled list with elbow problems.