CLEVELAND — The reigning AL Cy Young Award winner will end April without a victory.
It was Kluber's fifth start of the year, all of which have ended in a Cleveland loss. The Indians have the worst record in the AL at 6-12 and have dropped seven of their last 10.
After going 18-9 last year, Kluber (0-3) is winless with a 4.24 ERA.
"Sure, the way things are going is frustrating, but we can't let it snowball," Kluber said. "I don't focus as much on me, individually, as I do on the team as a whole. I know that you can't make up for what we've done in the first month in a couple of days, though."
Moustakas is 11 for 25 (.440) in his career against Kluber, who struck out five and walked a pair. The ace recorded his 500th strikeout in his 80th game, becoming the second-fastest Cleveland pitcher to do behind Herb Score's 68.
"There is never a question about Corey's effort, and I'll take him any day of the week," Indians manager Terry Francona said. "Frustration is there with everybody right now, but I know he'll stay after it. He's trying to get on a roll, and once he does, he'll feel a lot better."
Two of the Royals' runs were unearned as the Indians committed a season-high three errors, all by infielders on groundballs.
Kluber became the third reigning Cy Young winner to have his team lose his first five starts, joining Frank Viola in 1989 and David Price in 2011, according to information by the Indians from the Elias Sports Bureau.
Cleveland, which only totaled four hits off five Royals pitchers, fell to a major-league worst 1-5 at home.
"In an ideal world, we would've won more games than we have already, but I think we'll refrain from reading too much into just a handful of games," Indians general manager Chris Antonetti said. "I continue to believe in the roster we have, and we'll play better than we have so far."
Indians 2B Jason Kipnis fielded a routine grounder by Hosmer in the first, but his throw sailed approximately 10 feet to the right of 1B Carlos Santana and ended up behind home plate. The next Royals batter, Kendrys Morales, hit a grounder to Kipnis, who made a clean play and received a mock cheer from the crowd of 9,668.
Royals: OF Alex Rios (broken left hand) was examined by a specialist and will begin strengthening exercises. Rios was hit by a pitch on April 16.
Indians: OF Nick Swisher, who had surgery on both knees in August, hopes to rejoin the team next week. He'll continue his minor league rehab assignment with Triple-A Columbus on Tuesday. "It's been a long road," he said. "I just want this to be over so I can get back here with my guys."
Royals: RHP Jeremy Guthrie, who allowed three runs in the first inning against Minnesota in his last start, is 5-5 lifetime against Cleveland, the team that drafted him in 2002. Guthrie pitched for the Indians from 2004-06.
Indians: RHP Trevor Bauer, who was scratched Saturday because of food poisoning, will start against the Royals. He pitched seven shutout innings against the White Sox in his last start.