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KANSAS CITY, Mo. -- Trey Hillman thinks Zack Greinke should make room in his trophy case for the Cy Young Award.

Greinke bolstered his Cy Young credentials by holding Boston to two hits in six scoreless innings, leading the Royals to a 5-1 victory over the Boston Red Sox on Tuesday night.

"I am biased," the Royals manager said. "Part of that bias is I've seen all his starts. I have a very difficult time believing there is anybody better in the major leagues, that's as good a pitcher as Zack Greinke is in the year 2009."

Greinke is 3-0 with a 0.26 ERA in his past five starts, allowing 18 hits and one earned run in 35 innings. With the stingy stretch, Greinke has lowered his major league-leading ERA to 2.08.

Greinke (15-8) became the first Royals pitcher to win at least 15 games in a season since Paul Byrd won 17 in 2002. Byrd (1-2) was the losing pitcher Tuesday. Greinke has two starts left and knows the Cy Young could be at stake.

"You can't have a bad game," he said. "Any mistakes can cost you."

Greinke's record likely would be more impressive if the Royals gave him better run support. He has failed to earn a win in six starts in which he allowed zero or one run.

"He'd be a 20-game winner," Hillman said.

Greinke left his previous start after five scoreless innings at Detroit with a bruised elbow. He was struck by Miguel Cabrera's line drive in the fourth.

"There was no pain," Greinke said. "I was getting kinda tired in the sixth. I didn't feel crisp in the last inning. I thought I was pretty good, but it wasn't amazing. I thought my change up was as good as it's been all year."

Hillman decided 91 pitches was enough against the Red Sox "with what he had incurred in his last outing with the contusion."

"He was free and easy with his delivery," Hillman said. "In the sixth, I felt like he began a labor just a little bit. With what happened, it just didn't make any sense to push him even to 100 pitches. He pitched a heck of a game."

Dustin Pedroia doubled in the first and Victor Martinez singled in the sixth, extending his hitting streak to 21 games, for the only hits against Greinke. Greinke walked three and struck out five, bringing his season total to 229, which is second in the American League.

"Greinke has everything," Red Sox manager Terry Francona said. "He has velocity. You look up and you see a fastball that's 91 and then you see one that's 97. Then you see a slider, maybe the best slider in the game. I was impressed with his stuff."

Red Sox designated hitter David Ortiz endorsed Greinke for the Cy Young.

"Why not," Ortiz said. "He's got good numbers for it. If I could vote for the Cy Young Award, I might give one vote to him."

Red Sox slugger Jason Bay said Greinke has "Cy Young stuff."

"There are some other guys who are deserving, but he's probably one of the top two or three guys, no question," Bay said. "He definitely was attacking the zone. He's been doing that to a lot to of other people."

The Royals batted around in a five-run first, which included Alex Gordon's two-run, two-out single. The first five batters reached base off Byrd and all scored. Byrd walked Billy Butler and Mike Jacobs on eight straight balls. Jacobs walked with the bases loaded.

Callaspo, who had three hits and a walk, singled home Mitch Maier with the second run of the inning.

Byrd settled down after the first, going 6 2/3 innings and not allowing another run. He yielded five runs on nine hits and four walks. Byrd exited in the seventh with the bases loaded after a single and two walks, but Mark Teahen flied out to center to end the inning.

"The first inning wasn't very pretty, but after that he pitched great," Francona said. "Five in the first, that's a big hole to dig out of especially with Greinke pitching."

The Red Sox scored their run without a hit in the eighth off Roman Colon, who walked Jacoby Ellsbury and Pedroia. After a double steal, Ellsbury scored on Martinez's grounder. Ellsbury tops the majors with 64 stolen bases.

Joakim Soria was summoned for the final out after David Ortiz walked and Casey Kotchman reached on Butler's error. Soria needed just one pitch to retire Mike Lowell on a fly to center to log his 28th save in 31 opportunities.

Game notes

Teahen, who had missed the previous 11 games with lower back spasms, and LF David DeJesus, who left after one at-bat Saturday with a sore right ankle, returned to the Royals lineup Tuesday. ... The Royals are 12-3 in their past 15 games. ... Pedroia has a 14-game hitting streak, his longest of the season. ... Royals pitchers have struck out 1,084, one shy of the club record for a season.