Garcia and Cubs starter Carlos Silva engaged in a pitcher's duel through six innings, with both allowing a pair of runs. But after Silva was lifted for rookie Andrew Cashner (0-1), Konerko broke the tie with a towering solo blast to left, giving the White Sox their 11th straight win.
The victory streak is the longest for the White Sox since June 17-27, 1961.
Konerko's blast made a winner of J.J. Putz, who hurled the eighth inning. Matt Thornton collected his third save of the year with a scoreless ninth.
Garcia scattered eight hits over seven innings, issuing no walks and fanning five.
Aramis Ramirez hit a solo homer for the Cubs, who have lost seven of eight against their cross-town inter-league rivals dating to last year.