McAllister (1-0), called up from Class AAA Columbus before the game, allowed two earned runs (four overall) on six hits during the outing, walking one and fanning five.
It was the fifth career start for the 24-year-old right-hander.
Chisox starter Philip Humber (1-2) lasted only 2 1/3 innings, yielding nine hits and eight runs with two walks.
Humber has been tagged for 20 earned runs in three starts since his April 21 perfect game.
Alexei Ramirez went 2-for-4 with three RBI for the White Sox.
Google buys drone maker Titan Aerospace
Pistorius testifies he didn't consciously pull trigger when he shot girlfriend