Rios drove in all of Chicago's runs, adding a two-run shot in the sixth inning. The veteran outfielder is hitting .421 with eight home runs and 21 RBIs against the Twins this season.
"Take Rios out of the game and we're still playing," Twins manager Ron Gardenhire said. "But you can't take him out of the game because he took us out of it."
Both of Rios' blasts came off Twins right-hander P.J. Walters, who made his first Major League start since June 13.
Walters wasn't sharp early, allowing a leadoff single to Dewayne Wise and consecutive walks to Kevin Youkilis and Adam Dunn before Rios launched Walters' 1-1 slider over the left-field bullpen for his second career grand slam.
Walters rebounded to pitch quite well after the opening frame. He escaped a two-on, one-out jam in the second and allowed only one baserunner over the next three innings. He opened the sixth by walking Paul Konerko before Rios followed with his second homer to center field, ending the right-hander's day.
"I just got a few good pitches and I connected well, so that's all you can hope," Rios said. "Hit the ball hard and see what happens."
Walters (2-3, 6.05 ERA) struck out four but walked three batters, all of which scored.
"To me, that was a big part of the game, not so much the homers but the walks," Walters said. "If I don't walk those three guys ... that turns into a one-run game and maybe it plays out completely different."
A day after exploding for seven and 10 runs in separate innings, the Twins couldn't get a big hit with runners in scoring position.
White Sox right-hander Jake Peavy (10-10, 3.22) tossed six innings of one-run ball to earn the win, but wasn't dominant after retiring the first seven batters he faced.
The Twins loaded the bases with one out in the third, but managed to score only once, when Eduardo Escobar's sacrifice fly plated Ben Revere.
Minnesota also had runners on the corners with one out in the fourth, but Trevor Plouffe popped out and Matt Carson grounded to short to end the threat.
"We had some opportunities," Gardenhire said. "We put some men out there on him and we were one of those hits away for about four or five innings it looked like."
Carson plated the final run on a sacrifice fly in the final inning as the Twins fell for the ninth time in their last 11 games.
Although the rotation received plenty of help in Tuesday's 18-run performance, Minnesota starters once again unable to do their part during the three-game series. Samuel Deduno, Monday's starter, was the only one to go six innings, while Tuesday starter Scott Diamond and Walters each went five.
"These guys that are trying to win a spot in this rotation or get a look need to start getting deep into games for us," Gardenhire said. "We got about 16 innings out of our starters in this three-game series and we need more."
The series against the White Sox capped Minnesota's stretch of 20 games in 20 days, including a doubleheader on Sunday. Only four of those came at Target Field, where the Twins return Friday to begin a four-game set with the Indians.
"It didn't feel like we went home at all," Gardenhire said. "We've played 20 in a row. That's a lot of baseball. We're a little worn out, the bullpen's a little beat up. A day off tomorrow, hopefully that will get us going a little bit."