With the victory, the defending two-time AL champions cinched a winning season with their league-best 82nd victory, increased their AL West lead over Oakland to 5 1/2 games and won their sixth straight series.
Ian Kinsler opened the Rangers' 10th against closer Greg Holland with a triple off the top of the center-field wall, just beating Lorenzo Cain's throw to third. Michael Young singled to center, scoring Kinsler to break a 4-4 tie.
Cain came close to catching Kinsler's drive.
"I thought was going to catch it at the wall, but the wind just kept pushing it. I made a good effort, didn't come down with it," Cain said. "I try to make every play out there I can, came up a little short on that one."
Yost was holding his breath for a few moments.
"He did a great job of tracking it and getting on it and timed it just perfectly. The wind kind of picked up a little bit and just pushed it, but I don't think he missed it by more than a foot," Yost said.
After the ball caromed off the wall, Cain got off a long throw to third baseman Mike Moustakas and came within a flicker of nailing Kinsler.
"I thought that was really close, I wasn't expecting that at all," Holland said. "It's kind of a high bounce. If it bounces lower, Moose doesn't have to go so far to tag and he probably is out."
The Rangers edged Kansas City in their season series, five games to four. They had to overcome an early 3-0 Royals lead to do it.
Jeff Francoeur got the Royals on the board with a solo home run in the second inning off Rangers starter Scott Feldman. The blast to left field was Francoeur's first homer since Aug. 6 at Chicago and delighted the Frenchy Quarter Night section in the crowd of 15,332.
Feldman gave up two more runs in the third inning. Cain doubled to left and scored as Alex Gordon lined a triple just inside the right-field line. Gordon scored on a single by Salvador Perez, who extended his hitting streak to 11 games.
But home runs are a big part of the Rangers' game and they quickly had Feldman even at 3-3 in the fourth inning. Their explosion came after Royals starter Luke Hochevar retired the first 10 batters he faced.
"You can't make mistakes on 'em. They do a lot of damage with the home run," Yost said. "They're geared for it. They've got an attack-type offense where they look for quick strikes and, you make a mistake, they get 'em."
Young singled to left and Josh Hamilton pounded a home run over the right-field wall, his 39th this season. Adrian Beltre made it back-to-back bombs with a 418-foot drive to center field, his 30th homer.
Old stuff for the Rangers.
"You look across the league and your real good lineups are also veteran lineups, too," Hochevar said, "which makes it tough right on top of that, because they're playing the game right with you and they're thinking with you, and they're trying to get one step ahead of you."
And they often do. The Rangers broke the 3-3 tie on David Murphy's single and Geovany Soto's double in the seventh, Hochevar's last inning.
However, in the bottom half Eric Hosmer crashed a 417-foot home run to left center field off left-hander Michael Kirkman, who had just relieved Feldman. It was Hosmer's 13th homer.
"It was a tough lefty [but] 3-1 is a hitter's count," Hosmer said. "You don't want to miss that good fastball going by, so I just tell myself 'don't miss it' and [I'm] just trying to stay through it right there. I got a good pitch and hit it well."
The Royals finished the homestand with a 5-5 record, including six one-run games which they also split. They have a 20-16 record since July 31, keeping their outlook on the positive side.
"We're not here to be competitive, we're here to win," Holland said. "You can ask anybody in the clubhouse: We don't want to just play close games. But that's something we're getting better at, learning how to beat tough teams, winning close games. Tonight was just one of those nights where there's one bad pitch, there's a ball off the wall and you lose the game."
The Rangers, presumed to be playoff-bound again, are a tough force with which to contend.
"We know we can play. This is arguably the best team in baseball, and we played them tooth and nail all four games," Francoeur said.
Against the four first-place teams in the three AL divisions, New York, Baltimore, Chicago and Texas, the Royals have a composite record of 19-18 this year.
"We've been playing those teams well this year," Hosmer said. "When we get down to the teams that aren't in first, if we just go in and keep that same mentality, we could be pretty good."