Mike Moustakas and Jeff Francoeur each 3, in the third test of a game series.
Teaford (2-0), making the second start of his big-league career, gave up one run on four hits to pick up the win. He struck out four and walked two.
"It was big that I was able to keep it a one-run game and give them (the Royals hitters) a little more time to crack through like they have in the past week. They did it again," Teaford said.
Rios' solo homer to left in the second gave the White Sox an early lead.
After stranding runners on second and third in the third inning, Chicago left the bases loaded in the fourth.
The missed opportunities came back to haunt the White Sox, as Kansas City scored three times in the fourth. Moustakas and Salvador Perez led the frame with back-to-back singles. The runners moved up a base on a sacrifice bunt. Escobar cleared the bases with a triple to left. Gordon knocked in Escobar with a base hit.
Moustakas' two-run homer to right in the fifth made it a 5-1 game.
"They (the Royals hitters) are always in the count," Stewart said. "No matter what the count is, they seem to be taking good swings and battling. In the first few innings, they were putting pressure on me and ended up getting to me in the last two (4th and 5th innings). They did a good job."
The Royals tacked on another run in the sixth. Will Ohman gave up a double to Escobar and a single to Gordon. Cabrera followed with an RBI single to make it a five-run contest.
Chicago got two runs back in the seventh. Nathan Adcock gave up a single to Tyler Flowers and walked Brent Morel. The runners moved up a base on Gordon Beckham's groundout. Flowers scored on a wild pitch. Juan Pierre's RBI groundout cut the gap to 6-3.
Getz's RBI triple in the home seventh made it a 7-3 game.
Francoeur's three-run homer off Josh Kinney in the eighth made it 10-3.
The Royals lead the season series with the White Sox, 9-5...Moustakas, Cabrera and Perez had three hits apiece for Kansas City, which had 18 hits total...Chicago went 1-for-7 with runners in scoring position and stranded six.