Lester (1-2) baffled the Chisox on five hits over seven innings, issuing only one walk while striking out seven.
The Red Sox ace came into Saturday's contest with a 6.00 ERA over 24 innings.
Franklin Morales and Vicente Padilla combined for a scoreless eighth inning and Alfredo Aceves earned his fifth save with a 1-2-3 ninth for Boston, which clinched its four-game series against Chicago and extended its win streak to six overall.
Adrian Gonzales drove in the game's only run with a fourth-inning single.
Jake Peavy (3-1) took the hard-luck loss for the White Sox, yielding just the lone run in a complete-game effort.
The Chisox right-hander allowed four hits, walked one and fanned seven.
Celebrity Families of 2014 [PHOTOS]