The Rays' All-Star left fielder went 3 for 3 against Greinke (10-7) after having just one hit in 16 previous at-bats against him.
"He hit a fastball away, a slow curve and changeup down," Greinke said. "He helped make it look worse than it could have been."
Willy Aybar hit solo homers from both sides of the plate for the Rays, who won nine of 10 games in the season series with Kansas City. Aybar homered left-handed off Greinke in the fifth and went deep from the right side against Ron Mahay two innings later.
"I've had a lot of good luck," Aybar, who is 4 of 6 against Greinke, said through a translator.
Greinke allowed six runs and 10 hits over five innings, and is 0-4 in six starts since his last win on June 28 at Pittsburgh. He is 0-3 in five career games, including four starts, at Tropicana Field.
"Today, it was just mainly bad pitching to start the game off," Greinke said. "It really wasn't them the first couple innings. It was more getting myself in trouble."
Royals bench coach John Gibbons filled in for manager Trey Hillman, who left the team after Sunday's 4-1 win over the Rays to tend to a family matter. Hillman might return on Tuesday.
Aybar's second homer came in a four-run seventh that put Tampa Bay up 10-3. Jason Bartlett, hitting in the leadoff spot because B.J. Upton was rested, knocked in two runs on his third hit, a triple to right-center.
Rays manager Joe Maddon said he might tweak his regular lineup, which could include keeping Bartlett, who is hitting .337, at the top of the lineup. Aybar might get more at-bats at designated hitter with starter Pat Burrell struggling.
"I wouldn't mind it," Bartlett said. "We had a huge game. I had a good game. If that's what he wants to do, I'm more than happy to do that."
Carlos Pena's two-run single in the first put the Rays ahead 2-0.
Crawford extended the Rays' lead to 4-1 with a two-run double in the second. He also had an RBI single in the fourth.
The Royals loaded the bases with two outs in the fifth, but failed to score when Willie Bloomquist flew out to center on a 3-2 pitch.
Greinke struck out nine and walked three during his 105-pitch outing. Yunieksy Betancourt hit his first homer since May 16, a solo shot in the ninth.
Greinke said playing baseball indoors is different. "It doesn't feel as much like a baseball game when you're inside," he said. ... Royals 3B Alex Gordon, hitless in 11 at-bats in the first three games of the series, was out of the starting lineup. Gordon is 0 for 34 overall at Tropicana Field. ... Tampa Bay RHP Chad Bradford (lower back) is throwing off a bullpen mound. ... Royals RHP Gil Meche (lower back strain) was scheduled to make a rehab start Monday night with Triple-A Omaha. ... Tampa Bay released RHP Wade Townsend, who was taken eighth overall in the 2005 amateur draft. He has battled shoulder and elbow injuries throughout his pro career. ... Kansas City OF Josh Anderson, acquired from Detroit last Thursday, could join the team Wednesday. Anderson has been with his wife awaiting the birth of their first child.