"Garza threw the ball really well," manager Joe Maddon said. "Realistically, they should have scored maybe one run. That was not one of our better defensive games. It bothers us a lot. That's not us."
Kansas City got a run in each of the first three innings and made them stand up with its best pitching combination taking center stage. Behind Greinke, the Royals used hard-throwing Robinson Tejeda for the eighth. Tejeda fanned Dan Johnson with the tying run at third.
That brought on All-Star closer Joakim Soria, who rang up his 36th consecutive save and left the Rays and Yankees even again with a three-day weekend to decide the AL East -- though by having bested the Bombers head to head this season, Tampa Bay would claim the AL East if the two teams finish with the same regular-season record.
"We've got to swing the bats better," Maddon said. "Some days we hit, some days we don't. We faced a good pitcher tonight."
The Royals might have considered themselves fortunate to have any runs -- much less three -- after the third inning. Garza slipped trying to make a play on Gregor Blanco's dribbler to the right side leading off the first, and Blanco wound up coming around to score. Garza limped slightly after the misplay, but shook it off to resume pitching.
In the second, Jarrod Dyson produced an RBI triple with a fly ball over Carl Crawford's head. Crawford was playing extremely shallow under instructions from the bench.
"We moved [Crawford] in," Maddon said. "With two strikes on him right there, we thought if Dyson was going to move it to the left side against Garza, he probably wasn't going to hit it that far. That was our call."
In the third, Carlos Pena fielded a grounder at first and appeared to have plenty of time to throw out Mike Aviles breaking for the plate. But Pena made a wild throw.
The Rays received a solo homer from Crawford in the fourth and a sacrifice fly by Reid Brignac to close within 3-2. But they never could get over the hump against Greinke and Co.
"We've seen Greinke earlier in the year," right fielder Matt Joyce said. "He threw well then and he threw well again tonight. It was a good game, but it just didn't fall our way. So we have to come back and try to win some games and win this division. That's not the biggest thing for us, but it would be a nice bonus for what's ahead."
Garza, who allowed just five hits over seven innings, felt his outing was one he can use as a springboard into the playoffs.
"I'm right where I need to be," Garza said. "I went out there and attacked them. A couple of things didn't fall my way, but I got my breaking ball over for strikes and was able to repeat. So that's huge going into my next start."
Although the Rays have dropped their last two games against last-place teams, Garza expects the club to bounce back in strong fashion.
"We've got to win out, that's all there is to it," Garza said. "We don't need to look at what the Yankees are doing, we just need to win."
The Rays can at least take solace in the knowledge that the next three Royals starters don't have anywhere near the resume of Greinke.
While Tampa Bay will be trying to knock off Kansas City, it will hope for a little help from Boston against New York
"If that's the way it has to be, that's the way it has to be," Maddon said. "I really have a lot of faith in our guys. I believe we are going to play well [Friday]. And then the next day and the next day. I really feel you're going to see good baseball out of us over the rest of the weekend."