The latest brilliance from Greinke (15-8, 2.08 ERA) nullified a nice rebound by Byrd, who after giving up a five-spot in the first, didn't allow a run for the rest of his performance, which lasted 6 2/3 innings.
Ever the stand-up veteran, Byrd wasn't in the mood for moral victories.
"I'm frustrated. I'm not going to get any sleep," said Byrd. "I thought I had great stuff tonight. I just overthrew it a little bit in the first inning. I couldn't get the outs. Against Greinke, you don't want to be down, 5-0. It just deflates the team. I'm real frustrated with myself. I'm glad I came back and threw some innings and helped the bullpen out, but I'm not really here to eat innings. That's not my job. I'm here to get on the mound and give our team a chance to win. Tonight, I didn't do that, and it's a little frustrating."
It was the second consecutive loss by the Red Sox, who now must win the final two games to salvage a split of this four-game set against the 63-88 Royals.
Despite the loss, Boston's magic number for clinching a postseason berth is now six after Texas fell to Oakland, 9-1, on Tuesday.
Byrd was wobbly out of the gate. He gave up two singles and a walk in the first, immediately loading the bases with nobody out. The crafty righty normally has good control, but he walked Mike Jacobs to force in a run.
"So much of his game is not walking people," said Francona. "That's part of the reason he got in trouble. He created some baserunners, they got some hits. After that, he was really good. Better than we had seen in any of his starts."
Alberto Callaspo followed with an RBI single to right, and Mark Teahen drove home another run on a fielder's-choice grounder. Alex Gordon's two-out, two-run single made it 5-0.
"Everything fell apart," said Byrd. "Everything. If they hit the ball hard, it was just out of reach of our guys. If I jammed somebody, it fell in. If I threw a good pitch, it was just off the plate. You have innings like that, but I wanted to go out and help our team out a little bit."
In five starts for the Red Sox, Byrd is 1-2 with a 6.04 ERA. He is scheduled to take the ball next on Sunday at Yankee Stadium.
The Red Sox seemed to have their best chance against Greinke in the sixth. Victor Martinez extended his career-high hitting streak to 21 games with a single. With Martinez on first and two outs, David Ortiz hit a bullet that seemed headed for the gap in right-center field. But shortstop Yuniesky Betancourt, playing in a shift, timed his leap perfectly, snaring the line drive to keep Greinke's gem intact.
Ortiz, who is hitting .232 this season, is past the point of getting overly frustrated by a web gem. He feels he's been on the wrong end of a few of them this season.
"I guess it's part of my game now," quipped Ortiz.
Greinke's game, all year, has consisted of brilliance.
"It wasn't amazing," Greinke said. "But it was a good outing."
The Red Sox gave it a pretty high rating.
"You look up and see a fastball that's maybe 91 [mph]. Then you see one that's 97 up. Then you see a slider," Francona said.
It was the first and last time the Red Sox faced Greinke in 2009, and that was the only thing they had to be grateful about.
"He's one of the best pitchers in the league, and he was getting ahead of guys," said Red Sox left fielder Jason Bay. "Whether he swung early or not, he used quality pitches and was getting ahead. His numbers indicate that. He's been doing that all year."