On a so-so afternoon for the right-handed rock of the rotation, the Indians didn't help him out.
"He didn't have his command from the get-go. Every inning, they really made him work," manager Terry Francona said after the Indians were denied their first sweep of a road series this year and dropped six games behind Kansas City in the AL Central.
The Indians also fell into sixth place in the AL wild-card race, remaining 4 1-2 games out of the second spot.
"Any time you lose a ballgame it's disappointing, whether it be going for the sweep or the first game of the series. They all count the same in the long run," said Kluber, whose winning streak was stopped at six despite eight strikeouts over seven innings.
Kluber (13-7) walked four, matching his season high, while giving up six hits and three runs. This was his first loss since June 30 and the first start in which he allowed more than one run since July 19.
"The fact that he still kept it a low-scoring game and us in the ballgame for that long, credit to him," second baseman Jason Kipnis said. "That's what we've come to expect from him."
Over his previous six starts, Kluber gave up only four earned runs over 47 2-3 innings on 27 hits and six walks and did not allow a homer.
Phil Hughes (14-8) struck out eight over seven innings to forge a four-way tie for the AL lead in victories, with Rick Porcello, Max Scherzer and Scott Kazmir. Rookie Zach Walters homered for the third straight game for the Indians, but that was the only damage.
"He has that ability. Pretty good pitcher," Francona said.
Trevor Plouffe put the Twins back in front with a two-run double in the sixth.
"Just didn't quite have enough," Kluber said.
Hughes actually said he thought his repertoire was as bad as it's been in a while, but he was still able to improve to 4-0 with a 1.32 ERA in four August starts. He avoided a walk for the 14th time in his 26 turns for the Twins.
"He's got one of those sneaky fastballs. It's not going to shoot up on any radar, but it ... plays up a little bit and it's hard to get the barrel to," Kipnis said.
Vargas went deep for the fourth time in 19 games since he was promoted from the minors. The long ball ended a scoreless streak of 20-plus innings and 64 outs for the Twins since they tallied five runs with their first five at-bats to start the series Tuesday.
Glen Perkins pitched a perfect ninth for his 32nd save in 36 attempts for the Twins.
Walters has four home runs and six RBIs in nine games since joining the Indians in the trade with Washington that sent shortstop Asdrubal Cabrera to the Nationals. "I think he's kind of come as advertised," Francona said. "It's just nice to see it in person."
Indians: Catcher Yan Gomes was replaced in the sixth inning by Roberto Perez after being hit by a ball on the side of the mask. Gomes was evaluated for a concussion, but Francona said he felt "a lot better" afterward. "We obviously had to keep an eye on him," Francona said.
Indians: RHP Carlos Carrasco (5-4, 3.27 ERA) starts Friday to open a three-game home series against Houston. RHP Brad Peacock (3-8, 5.47 ERA) pitches for the Astros. Carrasco takes his third turn since rejoining the rotation, following three months in the bullpen. He's pitched 12 scoreless innings, with five hits and no walks allowed over the last two starts.
Twins: Host Detroit this weekend for four games in three days, including a doubleheader. RHP Tommy Milone (6-4, 3.99 ERA) starts following the shortest outing (only four outs, seven run) of his career.