Jack Hannahan's two-run double highlighted a three-run fifth, as Cleveland sent Kansas City to its 12th straight loss with a 4-3 decision in the opener of a three-game series.
Shelley Duncan and Carlos Santana knocked in a run each for the Indians, who have won four of five overall.
Derek Lowe (3-1) worked six strong innings, scattering eight hits and one run while fanning five. The veteran was able to claim victory despite his Florida home being burglarized earlier in the week, at the cost of his 2004 World Series ring and other losses totaling $90,000.
"This has obviously been a tough week," Lowe admitted. "It was nice to get back to work today. We played really good defense all around."
Chris Perez allowed an RBI hit from Billy Butler to pull KC within a run in the ninth, but after an Eric Hosmer single, retired Jeff Francoeur on a grounder to end the game and earn his seventh save.
Mitch Maier finished with three hits and two RBI for the Royals, who haven't won since April 10 at Oakland.
Jonathan Sanchez (1-1) lasted 4 2/3 frames, allowing four hits and four runs with a season-high seven walks.
Asdrubal Cabrera began the deciding frame with a walk, Santana also walked with one down and Travis Hafner's base on balls filled the sacks. Duncan lifted a fly to center which scored Cabrera then Hannahan stroked a double to right which brought in two more for a seemingly-safe 4-1 edge.
"It was a good pitch," said Sanchez of Hannahan's key hit. "I hit my spot there, but he just got after it."
But the visitors got one back in the eighth when Maier's double with runners on the corners and two out plated Hosmer, then made sure Perez worked through trouble in the final frame.
Chris Getz hit a leadoff double and moved to third on an Alez Gordon dribbler to the mound. He scored on Butler's grounder to short for a 4-3 game, but Francoeur's ball was snapped up by Hannahan, who flipped to second for the final out.
Cleveland picked up a run in the first as Jason Kipnis was plunked, moved to third when Cabrera followed with a double and came in on Santana's fly to left.
KC equalized in the fourth as Mike Moustakas singled with one out, moved to second on a wild pitch, reached third on a line out and scored on Maier's single.
Cleveland batters walked nine times in all...Sanchez was tagged for six hits and five runs over 2 2/3 innings in his first start vs. the Tribe on April 14 in Kansas City...Perez tied Baltimore's Jim Johnson atop the American League saves list...The Royals last lost as many as 12 in a row in 2008.