The homer was Rios' 24th, matching a career high set in 2007 with the Blue Jays. He's also just one RBI away from tying the career-high 88 he drove in two seasons ago.
"He's been huge for us," second baseman Gordon Beckham said of Rios. "You can look at his stats and say, 'Wow, that's awesome.' But he's been a big part of everything we're doing. He's had some huge hits for us. He's carried us parts of the year. We need him down the stretch. We need everybody down the stretch, but he's been awesome this whole year. He's a good teammate, and you cheer for him."
Beckham hit the second long ball of the night, a 1-0 blast off Hochevar to lead off the top of the sixth that knotted the game at 2.
"It was a fastball. He left it over the plate, and I was fortunate enough to get it," Beckham said. "Good time because we've struggled here a little bit. It's good to go tie that game in the sixth, and that way it's not getting down to the eighth and ninth with the guys they've got in their bullpen. I guess it was good timing."
The first of the three solo shots came on the game's second pitch, as Alejandro De Aza connected on his third leadoff home run of the season to give the White Sox, who improved to just 6-10 against the Royals, an early lead.
De Aza returned to the lineup Tuesday night after struggles against the Twins sent him to the bench the previous two days.
"I was having some tough times. I started working a little bit with the hitting coach and tried to get my approach right," De Aza said. "Today I felt much better, and I was successful."
It was good timing by manager Robin Ventura to return De Aza to the lineup.
"I know what I'm doing," Ventura said wryly.
While the trio of homers was good enough on Tuesday, the White Sox were focused on not making that the norm.
"Those are runs that happen every once in a while where you don't get enough runs doing the little things and you depend on long balls," Rios said. "Hopefully, we don't have to depend on those for too long. The last couple of games we got big hits. Instead of hitting bombs, we got big hits and scored runs on those hits, too. Hopefully, we can make a good combination of defense and offense and keep winning games."
The White Sox got a good combination of offense, defense and starting pitching, as Gavin Floyd was solid in his second start since returning from the disabled list. He allowed two runs to score on a Billy Butler single in the first inning, but he settled down after that.
He allowed just the two runs on five hits over seven innings. He picked up his 10th win -- his first since Aug. 15 -- in his longest outing since July 7 against the Blue Jays. It was a bounce-back performance from his last start, when he lasted just 4 2/3 innings. He now has won double-digit games in five straight seasons.
"The first inning didn't exactly go the way I wanted it to, but it is what it is and I just keep on trying to make pitches. It was a rough first, especially after getting a run. You want to shut them down," Floyd said. "I felt good out there, and I just wanted to execute pitches and let the defense work."
"That first inning, they got him for a couple, but other than that he was sharp. I think he was getting ahead. He had just great feel for pitches," Ventura said. "I almost wanted to leave him in there a little longer, but we've got guys coming in to pick him up."
Floyd only threw 78 pitches on the night before yielding to Matt Thornton and Addison Reed, who combined to throw two scoreless innings, with Reed earning his 28th save.
After Royals outfielder Jeff Francoeur doubled with two outs in the fourth, Floyd, Thornton and Reed retired the final 16 Royals in a row.
The win extended the White Sox winning streak to five games, their longest since winning six straight Aug. 20-26.
With just 15 games remaining, the White Sox aren't simply counting down the days until the season is over.
"I think the finish line is close. But we can't really focus on anything like that," Beckham said. "We've just got to focus day by day going out there and making sure that we get prepared enough to go out there and hopefully win a ballgame. That's all that should matter and all we should be thinking about."