It was the second run of the day for the A's, who got an RBI base hit from Eric Sogard in the bottom of the third against starter Erik Johnson, following Stephen Vogt's leadoff double. Vogt, battling for a catching job, is 9-for-18 (.500) in seven games this spring.
Johnson went four innings for the White Sox, giving up one run and four hits with one walk and five strikeouts.
For the A's, righty Dan Straily walked three in 3 1/3 innings, but didn't give up a run, allowing two hits and fanning two along the way.
The White Sox scored an unearned run off lefty Tommy Milone in the sixth. Gordon Beckham ended up on third after Yoenis Cespedes, battling the sun, couldn't catch a routine fly ball, and pinch-runner Micah Johnson proceeded to come home on a Jeff Keppinger groundout.
Johnson scored the tying run in the eighth when A's second baseman Jake Elmore couldn't handle a grounder by Blake Tekotte.
Oakland's Brandon Moss collected two hits, bringing his Cactus League average to .533.
Up next: Jarrod Parker makes his third start of the spring on Monday, when the A's travel to Glendale, Ariz., to take on the Dodgers for a 1:05 p.m. PT start in a free exclusive audio webcast on athletics.com. The game will also air on MLB.TV.
The right-handed Parker had a shaky Cactus League debut, but followed it up with an impressive outing against the Brewers, allowing one run on two hits in 3 1/3 innings. He's considered one of three pitchers manager Bob Melvin is considering for the Opening Day nod. Alberto Callaspo is expected to make his second start at first base.
Jane Lee is a reporter for MLB.com. Read her blog, Major Lee-ague, and follow her on Twitter @JaneMLB. This story was not subject to the approval of Major League Baseball or its clubs.