Feldman (6-6) worked into the eighth inning and became the first pitcher in 29 years to win six consecutive decisions after starting the season 0-6. Doyle Alexander did it in 1983 with the New York Yankees and Toronto Blue Jays.
"I've just sort of gone about my business and felt the same way the whole year," Feldman said. "It's just nice to get an opportunity to go in there and help out and keep my focus on that, making good pitches."
"It's been very timely," manager Ron Washington said. "He's been our security blanket, really. Things go wrong in the rotation and we've been able to just plug him and he eats up innings. He's executing his pitches really well."
Mike Olt, called up Thursday from Double-A, added two RBIs for the AL West leaders.
"It's like the '09 Feldman," Hamilton said. "He's throwing the ball good, mixing his pitches up well. He's getting the job done. He's keeping the ball down. He's not throwing it over the middle of the plate. Everything has got so much movement on it."
Feldman, who was 3-0 with a 1.96 ERA in July, held Kansas City to six hits and two runs over 7 2/3 innings. The Royals, who lost for the 23rd time in 32 games, scored twice in the fifth on groundouts by Chris Getz and Alex Gordon.
"We did a pretty good job against him," Kansas City manager Ned Yost said. "He had a tremendous two-seamer, a tremendous splitter and a tremendous curve. He was really hard to hit and I think we did a good job of getting some hits off him."
Olt, who had 28 home runs and 82 RBIs with Double-A Frisco before his promotion, hit a sacrifice fly in the fourth and an RBI single with two outs in the sixth.
"A big sac fly, a big hit up the middle with two outs," Washington said. "That was huge. That gave us a little bit more of a cushion."
Royals rookie Will Smith (2-4) gave up four runs and eight hits over 5 2/3 innings in losing his third straight start.
The Rangers could have scored more runs in the third if not for a gaffe on the bases.
Hamilton stopped at second on Adrian Beltre's single before Nelson Cruz's grounder caromed off the glove of shortstop Alcides Escobar for an error. The Royals, however, converted that into a double play when Beltre was caught between second and third and tagged out. Cruz attempted to go to second on the play and was out to end the inning.
"We could have put more on the board in the third inning, but it just wasn't to be," Washington said.
The Royals fired first base coach Doug Sisson and replaced him with Rusty Kuntz, who served in that capacity from 2008-10. "We felt a change was needed," Yost said. ... Nathan was unavailable again Saturday after throwing 62 pitches in two innings on Wednesday and Thursday. "That's a lot of pitches," he said. "I'm trying to get rid of some normal tightness and get back to throwing more free and easy. Two nights of breather will loosen me up." ... LF David Murphy remained in the Texas lineup, although the Royals started a left-hander. "He's hot," Washington said. "I'm going to ride him. At some point you have to take advantage of what guys are doing when they're hot." The left-handed hitting Murphy entered batting.378 against left-handers. ... Feldman has not pitched a complete game in 93 career starts, the third-longest active streak in the majors. ... LHP Derek Holland will make his sixth start Sunday for the Rangers since coming off the disabled list from shoulder fatigue. The Royals will start RHP Luke Hochevar, who is 2-0 with a 3.08 ERA in four career starts against the Rangers.