They had only scored more than four runs once in their first seven games, lost 1-0 to Tampa Bay the previous night, and were the last team in the majors without a home run.
"The offense gave us a lot of breathing room," Royals manager Ned Yost said. "Runs are nice. Home runs are nice too. But I don't care how we get them."
Jeremy Guthrie (2-0) recovered from a sloppy start to hold the Rays to four hits over seven innings. The only run he allowed came on Desmond Jennings' homer in the fourth.
The right-hander, who turned 35 on Tuesday, was coming off a rough start against the White Sox. But Guthrie navigated trouble in each of the first three innings, leaving five Rays on base, and then retired his final 12 batters to hand a 7-1 lead to his bullpen.
"You're going to make good pitches and they're going to hit them hard sometimes," Guthrie said. "But if you make good pitches, more often you're going to get an out."
Jake Odorizzi (1-1) mowed through the Kansas City lineup the first three innings before allowing Butler's RBI groundout and Gordon's run-scoring single in the fourth.
Everything unraveled for him in the fifth.
Lorenzo Cain singled off Odorizzi to lead off the inning, and Aoki followed two batters later with a triple to right. Giavotella, recalled from Triple-A Omaha to replace injured second baseman Omar Infante, hit a sacrifice fly to make it 4-1.
Hosmer and Butler followed with back-to-back singles, and Gordon popped a pitch to right-center that hung up long enough in the wind to land over the fence.
"I was making my pitches, getting what we wanted. Balls were on the ground, not hit very hard, they just didn't find anybody," Odorizzi said. "That's baseball."
It was a rough way for Odorizzi to return to Kauffman Stadium, where he made his big league debut with the Royals in 2012. He allowed all seven runs on 10 hits and a walk in five innings.
"He didn't do anything wrong today. He threw really well," Rays manager Joe Maddon said. "Those numbers are highly deceiving. Aoki's ball was well-struck and the home run was part of natural forces of nature. People who look at that line are going to misinterpret that outing because he threw the ball extremely well. It was just an unfortunate game for him."
The Royals squandered a scoring chance with runners on first and second and one out in the second inning when Mike Moustakas struck out and Gordon was thrown out heading to third.
Royals manager Ned Yost trundled onto the field and challenged the call. After a review of 2 minutes, 10 seconds, the ruling made by third base umpire Quinn Wolcott stood.
It hardly mattered the way the Royals were swinging -- and the way Guthrie was pitching.
"They've played really well against us the last couple of years. You've got to give them credit," Maddon said. "Just keep doing that stuff and it will come back to us."
Kelvin Herrera gave up two runs in the ninth for KC. ... The Royals activated RHP Louis Coleman (bruised finger) from the DL and optioned RHP Aaron Brooks and LHP Donnie Joseph to Triple-A Omaha prior to the game. ... The Rays optioned LHP Jeff Beliveau to Triple-A Durham after the game. ... The Rays visit Cincinnati this weekend. They've never won at Great American Ball Park. ... Kansas City plays 13 of its next 16 on the road beginning Friday night in Minnesota.
Copyright by STATS LLC and The Associated Press
KANSAS CITY, MO - APRIL 09: Starting pitcher Jeremy Guthrie #11 of the Kansas City Royals pitches during the game...