Salvador Perez also drove in three runs with three hits for the Royals, including a solo home run off Cueto in the second inning. For the Reds, Ryan Ludwick and Todd Frazier hit back-to-back homers in the first inning against starter Danny Duffy.
The ball was flying.
"You gotta love Arizona," Hosmer said, laughing.
It wasn't pure power?
"Maybe about 30 percent, and 70 percent Arizona," he said, then added more seriously: "I got a good piece of it. Arizona might have helped it out a little bit, but you're not looking for results, you're just looking for good swings out here. You want to work good at-bats, and I think as a team we did that today so it was productive."
Cueto couldn't get through the third as the Royals put up a five-spot on six hits.
After singles by Justin Maxwell and Jarrod Dyson, Hosmer belted a homer to center field. Billy Butler and Alex Gordon followed with back-to-back triples into right field and Perez upped the Royals' lead to 6-4 with a double. That ended Cueto's outing after 58 pitches.
Perez drove in a third run with a fifth-inning single.
All that came after the Reds ganged up on Duffy for four runs in the first inning. Chris Heisey led off with a double and, two outs later, Ludwick hammered his first home run to left field. Frazier followed with a wall-clearing drive to center. Devin Mesoraco singled and got around on two errors.
"We had a little energy today," Frazier said. "Everybody pretty much got in the box and tried to get better. Everybody felt pretty good. It was fun to watch that kind of stuff. You can build on that. As a guy on deck or a guy in the dugout, you want to build on something like that."
Bruce Chen took over in the fourth for the Royals and reeled off three perfect innings with three strikeouts.
Up next for the Royals: Omar Infante, who's missed the last five games because of shoulder stiffness, is scheduled to get back in action for the Royals on Wednesday, but not at second base. He's expected to be the designated hitter against the host Diamondbacks at 2:05 p.m. CT at Salt River Fields at Talking Stick. James Shields will make his second start and plans to expand his 10-pitch, one-inning first outing to about 45 pitches. Also scheduled to pitch for the Royals are Chris Dwyer, Tim Collins, Francisley Bueno, Aaron Brooks, Jason Adam and Scott Alexander. For the D-backs, left-hander Patrick Corbin is the probable starter. This will be the Royals' first exposure to the expanded instant replay system.
Dick Kaegel is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.