"I think it's just the natural progression of a real good pitcher," Boston manager Terry Francona said, flashing a slight smile.
Buchholz (7-3) allowed four hits -- three singles -- walked four and struck out four in posting his sixth win in seven starts. He's 3-0 with a 1.32 ERA in his last three outings.
With his name popping up in trade rumors as the club looked for a middle-of-the-order hitter last season, Buchholz bounced between Triple-A Pawtucket and Boston.
On Saturday, he helped the Red Sox rebound after consecutive losses.
"No added stress," he said of taking the mound with the team on a losing skid. "I just felt like going out and being confident and pitching is good enough."
The Royals entered the night leading the majors with a .282 team average. But they had trouble solving Buchholz's mixture of offspeed pitches that featured a good changeup and curveball to go along with a hard fastball.
"We matched up against a guy who was on top of his game and is one of the best pitchers in the American League," Kansas City manager Ned Yost said. "He didn't make many mistakes."
The Red Sox had lost the first two of a four-game series after sweeping the majors' best team, Tampa Bay, earlier this week.
The loss snapped the Royals' three-game winning streak.
Mike Lowell drove in Boston's lone run with an RBI grounder.
"He helps you win. He always does," Francona said. "We talk about how good he is defensively, but with the game on the line he's even better."
Greinke (1-6) was coming off his worst start in nearly two years, when he was pounded for eight runs -- seven earned -- in 3 1/3 innings last Sunday against Colorado. He gave up the one run and five hits, walked three and struck out three in six innings.
"It was a battle all day. It wasn't a smooth six-inning game," he said. "It was a real stressful game. It'll take a toll if every game is like that."
"Given the situation, that's about as big of a play that you can make look routine," Bard said.
Jason Kendall doubled leading off the eighth against Bard and advanced on a sacrifice. After Mike Aviles fanned on a slider outside the strike zone, Pedroia made his play on a sharply hit ball to his right when he slid to one knee, got up and fired to first.
Buchholz allowed only one baserunner to third, coming in the fourth when Aviles advanced from second on Billy Butler's double-play grounder. The right-hander then retired Jose Guillen on a grounder to first.
In the fifth, Boston loaded the bases with two outs on a double by Ortiz, intentional walk to Martinez and walk to Beltre, but Drew bounced to first.
- Francona rested regular 1B Kevin Youkilis and inserted Lowell into the lineup. He said the move was made easier due to Lowell's good numbers (4-for-7) against Greinke.
- Francona also said that the team will "slow down" right-hander Josh Beckett, who is on the 15-day disabled list after hurting his back on a slippery mound in Yankee Stadium on May 18. "He's not able to complete his delivery consistently enough," Francona said. "We don't want it to lead to anything else." Beckett had thrown a side session Friday.
- The Royals placed RHP Gil Meche on the 15-day DL due to shoulder bursitis and recalled RHP Victor Marte from Triple-A Omaha. It's the second DL stint for Meche, 0-4 with a 6.66 ERA, due to the same injury.
- Royals reliever Kyle Farnsworth went unnoticed as he walked in shorts and a T-shirt with his son through the crowd in the lower concourse about an hour before the game.