Teaford, a 27-year-old, debuted this season as a reliever for the Royals and appeared in 23 consecutive games out of the bullpen before getting a start on Sept. 11 in Seattle.
He tossed five scoreless innings in a 2-1 Kansas City victory, then got win No. 2 after allowing one run on four hits in six innings of a 10-3 defeat of the White Sox on Sept. 17 in Kansas City.
He walked two and struck out four in the outing, lowering his earned run average as a starter to 0.81. His combined ERA for the season is 2.54 in 39 innings.
Bruce Chen (12-8) allowed one run on two hits and a walk through eight innings in the Royals' ninth win in 11 games.
"He made some adjustments and threw a very, very good ball game," Royals manager Ned Yost said.
Zach Stewart (2-6) allowed nine runs on 12 hits in four-plus innings as Chicago lost for the 10th time in 13 games.
"They're a good hitting team and they're not a selfish team," Stewart said of the Royals.
The White Sox reply with lefty John Danks, who's undefeated in 11 career starts against Kansas City, including a win on Sept. 18.
In that game, a 10-5 triumph at the Royals' Kauffman Stadium, Danks scattered 10 hits and allowed three earned runs in six innings.
He also beat the Royals on Aug. 14 with six innings of four-hit, two-run ball for a 6-2 victory.
This will be the 150th big-league start for Danks, who's 4-4 this season in 13 outings at home.
Kansas City is 10-6 versus Chicago this season.