They got a lift by winning the homestand opener, 8-3, over the Indians.
So when Mendoza's father Alfredo and uncle Jose Luis came in from Veracruz, Mexico, to join his wife Monica in the stands, the right-hander was primed for a big night and he gave it to them.
"That's a good motivation," he said. "I've got to do well with my family in town so I'm happy with it."
An efficient Mendoza worked through 94 pitches on the sultry night and was given a standing ovation by the 17,033 fans when he was taken out. He gave up just four hits and two runs. He struck out three and walked one.
Indians manager Manny Acta credited Mendoza's curveball for much of his success.
"We've got a lot of lefties in our lineup, so he backdoored that and busted them in with that fastball," Acta said. "He had a good two-seam fastball coming back over the plate. We're struggling, but he pitched well."
The last batter Mendoza faced, Jack Hannahan, hit a one-out double. Left-hander Jose Mijares took the mound and Shin-Soo Choo was hit by a pitch although Royals manager Ned Yost questioned home-plate umpire Mike Everitt about the call. Asdrubal Cabrera singled to left, scoring Hannahan before Mijares got the last two outs.
The right-handed Mendoza was working against an Indians lineup with eight left-handed hitters. So it was a natural move to bring in Mijares.
"With all the lefties, we were going to maximize our matchups," Yost said. "We went with Mijares and he was able to get us out of the inning with a three-run lead."
Greg Holland, in his first save opportunity since closer Jonathan Broxton was traded, zipped through a one-two-three Indians ninth for his first save this season.
"I was kind of amped up today. I kind of had to calm myself down, tell myself it was just another three-run game," Holland said. "Fortunately, you get that three-run lead, you can just attack the strike zone. As long as you stay ahead, your team's going to win the game for the most part. I had to calm myself down a little bit, but after I got strike one on the first hitter, it was just kind of another day at the yard."
Four stolen bases in the first two innings helped the Royals spring ahead, 4-0.
A double steal -- Alcides Escobar to third and Lorenzo Cain to second -- ignited a two-run first inning against Indians right-hander Zach McAllister.
"I did that on my own. I saw the pitcher had a high leg kick. Billy [Butler] was hitting and I knew he wasn't going to throw many fastballs to Billy," Escobar said. "And I said, 'I'm going, I've got a chance to get to third base.' "
Escobar not only got to third base, he scored on catcher Jose Lopez's wild throw. That moved Cain to third base and he scored as Butler bounced out to the right side.
Two runs in the second inning came after two outs, again with aggressive baserunning. Chris Getz walked and stole second base, scoring on Jarrod Dyson's single. Dyson swiped second and sprinted home on Alex Gordon's double to left field.
"That's our goal: use our legs," Dyson said. "Me, Alcides, Getz and Cain, we can run pretty good. We're never going to let our legs go into a slump. If we can get a bag, we're going to try it."
The Indians got a home run by Carlos Santana in the fourth inning but the solo shot was given a video review before it became official. The drive off Mendoza's 3-1 pitch just cleared the right-center field wall, but the ball immediately bounced back onto the field and Yost wanted to be sure. So the umpires obliged with a quick look on a TV underneath the stands. No doubt about it, a homer, Santana's 10th.
The Royals added a two-out run in the fifth inning on Escobar's single to right and Cain's double to left field for a 5-1 lead and two outstanding defensive plays helped Mendoza hold it.
Escobar raced into left field for a catch of Michael Brantley's floater in the sixth inning. After Santana walked to start the seventh, third baseman Mike Moustakas dug out Lopez's shot and started a lightning double play.
"That's all Getzy, though," Moustakas said. "All I was trying to do was get it over to him as fast as I can. I didn't make a good throw at all and he handled it and was able to turn it. It was real impressive."
The two-victory surge was modest but enough to get the Royals stoked up.
"After two wins in a row, everybody's going, 'C'mon, let's go, play hard,'" Escobar said. "And that's better, you know."
And maybe the boys were just loosened up by that Midwest heat.
"It could be, it's been awful hot out there," Yost said. "Normally it'll cool off when the sun goes down, but it didn't cool off. As long as we're winning games, I don't care what it's like out there."