"I felt absolutely horrible warming up in the bullpen," Meche said. "It's weird. I just went out and tried to attack the [strike] zone and things fell in place."
Alberto Callaspo had a career-high four hits including his first career grand slam for Kansas City. The Royals also received two-run homers from Jose Guillen and Miguel Olivo to win for only the fourth time in their last 19 games.
"It's been a while," Guillen said of the offensive outburst. "Hopefully we learned something and will start playing real baseball again."
Kansas City averaged only 2.4 runs and was shut out four times in its previous 16 games.
Meche (3-5) yielded four hits and three walks in his first win in seven starts. Since beating Minnesota on May 3, the right-hander had gone 0-3 with a 4.02 ERA with the Royals losing each time. Meche improved to 19-6 in his career in the month of June. The 30-year-old has only a 62-67 record in all other months combined.
• Indians: Sizemore to have MRI
Guillen and Olivo each hit their seventh homers of the season in the Royals' four-run fourth off Carl Pavano (6-5). Callaspo connected off the right-hander to make it 9-0 in the fifth.
Pavano, coming off a 6-0 win in Chicago over the White Sox on Friday, yielded nine runs and 11 hits over 4 2/3 innings. He had gone 6-1 in his previous eight starts since May 1.
The right-hander blamed himself for losing focus after not getting off the mound quickly enough to cover first base on a ground single to the right side by Mark Teahen that loaded the bases for Callaspo.
"I didn't do my job," Pavano said. "That was a mental error that can't happen. Then I gave up the grand slam. That was pretty much the ballgame."
Cleveland dropped back into last place, a half-game behind the Royals in the AL Central. Kansas City, which led the division by three games on May 7 has gone a major league-worst 7-22 since.
"Through this whole nightmarish stretch, the good thing is we still played hard," Royals manager Trey Hillman said. "The results tonight were much better. We scored six of the nine with two outs. That's always good to see."
Sharp baserunning helped the Royals take a 1-0 lead in the second. Teahen drew a walk and was off with the pitch when Callaspo hit a line drive to right. Teahen reached third on the single, then scored on a sacrifice fly by Brayan Pena.
The Indians' best chance against Meche came in the fifth. They loaded the bases on two singles and a walk, but center fielder Willie Bloomquist made a nice running catch in right-center of Jamey Carroll's bid for extra bases to end the inning.
"It was good to see Gil get out of the bases-loaded jam," Hillman said. "That was a real confidence booster."
Cleveland right fielder Shin-Soo Choo went 0 for 3 with two strikeouts against Meche.
"He didn't throw anything down the middle," Choo said. "Everything was on the outside corner or the inside corner or in the dirt. He was throwing mostly fastballs and had a great change-up. He was pretty good."
- The Indians didn't come close to matching what Cleveland OF Rocky Colavito did exactly 50 years ago on June 10. "The Rock" went 4 for 4 -- all home runs, with six RBI in an 11-8 win at Baltimore. Colavito became the eighth of 15 players in history to hit four homers in a game.
- Cleveland has been shut out four times -- all at home, twice by Detroit and twice by the Royals.
- Indians C Kelly Shoppach fanned twice and has hit .133 (4 for 30) over his last nine games.
- Royals OF Coco Crisp didn't start against Pavano because his sore right shoulder bothers him hitting left-handed. The switch-hitter played Tuesday against LHP Cliff Lee and is expected to start Thursday against LHP Jeremy Sowers.
- Indians OF Grady Sizemore, on the disabled list since May 31 with an inflamed left elbow, is scheduled for an MRI on Monday.