With a win, Minnesota could have pulled even with Detroit atop the AL Central standings. The Tigers lost 7-4 to Toronto.
Detroit and Minnesota already have clinched playoff berths, but the division champion still has to be decided. If they finish tied, the Tigers would take the division title because they won the season series 11-8. The Twins remained one game behind Detroit with four to play.
"It's disappointing," Twins right fielder Michael Cuddyer said. "We feel that we missed an opportunity. But at the same time, we can't get down. We haven't gotten down all year."
Redman (11-10) allowed three runs and nine hits for the Royals, who won for the first time since a 7-4 home victory Sept. 16 against Seattle. John Buck went 2-for-4 with two RBI.
Joe Nelson allowed a leadoff homer in the ninth to Cuddyer but earned his ninth save.
The victory was the first for Kansas City interim manager Billy Doran (1-5), who is filling in for Buddy Bell. He is still recovering after having his tonsils removed Friday.
"It was nice to get a win for Buddy," Doran said. "I know he's home watching. As tough as it is on us, having to go through all this, I don't think words can describe what an emotional time this has been."
Rondell White was 2-for-3 with two RBI for Minnesota, which had won four in a row.
Carlos Silva (10-15) lasted just four innings for the Twins and might have cost himself a chance at starting in the postseason. The right-hander gave up five runs and six hits, including the two home runs. Silva has allowed 38 homers this season, worst in the major leagues.
"I felt good, but I missed my spots," Silva said. "No matter what team you face, if you miss your spots, you're going to get hurt."
Manager Ron Gardenhire said before the game that the Twins will use four starters for the first round of the playoffs, though it's unclear who will follow Johan Santana and Boof Bonser in the rotation.
Silva is one of the candidates to start against the Yankees or Athletics, but he has pitched poorly of late. He gave up five runs in 5 2/3 innings in his previous start, a 7-3 loss to Baltimore.
Silva, a workhorse in Minnesota's rotation last season, struggled early this year and was demoted to the bullpen for three weeks after going 2-6 with an 8.80 ERA through eight starts.
The sinkerballer returned to the rotation in June and steadily improved in the second half, making three consecutive quality starts earlier this month, but he has been hit hard his last two games.
"I don't feel like I've helped the team this year," Silva said. "I don't feel like myself. I've pitched two good games, then two bad games. I haven't been consistent.
"I don't expect anything for the playoffs. I don't think I've pitched good enough. But if I get the opportunity, I'm going to give 100 percent."
Despite Silva's recent struggles, Gardenhire said it's possible the big righty, who threw just 54 pitches Wednesday, could come back on three days' rest and pitch Sunday's season finale against the Chicago White Sox.
"If we need to win, it probably will be Carlos, and he'll have to suck it up," said Gardenhire, who added that he wasn't even thinking about deciding on his third and fourth starters for the first round.
For Redman, it was his second consecutive strong outing against the Twins. The left-hander, who broke into the major leagues with Minnesota and pitched for the Twins from 1999-01, threw a five-hit shutout in Kansas City's 2-0 win Aug. 29 at the Metrodome.
"My fastball's not any slower or faster, it's just moving a little better and I was able to control it there," said Redman, who was coming off the worst start of his big league career, when he allowed nine runs and seven hits in one-third of an inning in Kansas City's 15-4 loss to Detroit on Saturday.
The Royals took control Wednesday with four runs in the fourth inning. Sweeney led off with a line-drive homer to left on an 0-2 pitch. Shane Costa and Buck added RBI doubles, Angel Berroa had an RBI single and Angel Sanchez hit a sacrifice fly.
"They got some runs, but we played good baseball today," Redman said. "They gave me an early lead, and it helped."
Buck added an RBI single in the fifth to give Kansas City a 6-1 lead. White's two-run single with two outs in the eighth brought the Twins to 6-3.
DeJesus' homer into the right-field upper deck in the third tied the score at 1-all. The Twins grabbed a 1-0 lead in the first inning on Joe Mauer's double.
- Redman's 11 wins are the most for a Royals pitcher since Paul Byrd won 17 games in 2002.
- Twins CF Torii Hunter went 0-for-3, snapping his 10-game hitting streak. He left the game after fouling a ball off his left big toe during a seventh-inning strikeout.
- Mauer went 2-for-4 to increase his major league-best batting average to .350.