Danks (6-6) gave up the four hits in seven innings with four walks and five strikeouts. Linebrink struck out four of the six batters he faced.
One of three hits by Alexei Ramirez was a third-inning, two-run homer.
Jermaine Dye had two hits and one RBI.
Chris Getz had two hits and stole home in the sixth. Dewayne Wise hit an infield RBI in the fourth for the White Sox, who took two contests in the three-game series.
Alfonso Soriano had two of the four hits for the Cubs, who have lost six of seven.
Losing pitcher Carlos Zambrano (4-3) surrendered five runs -- four earned -- and nine hits in 5 1/3 innings.
The Windy City rivals split six meetings in this season's inter-league play.
Notable deaths of 2014 [PHOTOS]