Carlos Santana's seventh-inning homer proved to be the difference, while Chris Perez retired Alex Gordon with the bases loaded in the bottom of the ninth as Cleveland nipped Kansas City, 4-3, in the opener of a three-game series.
Asdrubal Cabrera and Yan Gomes also went deep for the Indians, who have taken five of their last six to remain alive in the AL wild card race. Cleveland trails Tampa Bay by 1 1/2 games for the final spot.
Ubaldo Jimenez (11-9) scattered seven hits and one unearned run over seven innings, striking out 10.
"His slider was really good," Royals manager Ned Yost said of Jimenez. "He was tough today."
Perez made things interesting before recording his 23rd save.
Gordon clubbed a two-tun homer in the eighth and Eric Hosmer added an RBI single for the Royals, who lost for just the third time in their last nine.
Ervin Santana (8-9) was tagged in defeat for four hits and four runs -- three earned -- while fanning seven in seven full frames.
Alcides Escobar doubled off Cody Allen to start the eighth and Gordon pulled the visitors within 4-3 on his homer to right.
In the ninth, Salvador Perez hit a leadoff single, Mike Moustakas walked and both runners were sacrificed into scoring position. Carlos Pena then fanned looking, but pinch-hitter George Kottaras drew a base on balls to fill the sacks.
That set up the confrontation with Gordon, which the Cleveland closer won by inducing a game-ending fly out to center.
"This is what you play for...to get into the postseason," Jimenez said. "Every game counts."
Cabrera's solo homer in the second put the Tribe on the board.
Kansas City's Jarrod Dyson hit a ground-rule double to kick off the third, then moved up on a groundout, but was erased at home on a grounder from Gordon.
Cleveland picked up another run in the home half, when Jose Ramirez hit a leadoff single and came all the way home when Hosmer -- after retiring Drew Stubbs on a grounder -- had his attempt to throw him out at third sail into left field.
Gomes' opposite-field smash made it 3-0 in the fifth, and the Royals finally countered on an RBI single from Hosmer in the sixth.
Santana went deep in the home half to restore the Indians' three-run edge.
Cleveland has won nine of 14 in the season series and taken six of seven at home ... Gordon's blast was his 100th career homer ... Gordon, Hosmer, Perez and Dyson accounted for eight of Kansas City's 10 hits, collecting two each ... The Royals opened September with a 6-2 mark, the hottest start to the month since the 1985 World Series championship club started 7-1 en route to a 22-14 finish that included an 18-12 September.