Ross, one of the A's hurlers competing to fill three open rotation spots, hurled three scoreless innings with two strikeouts and two walks.
"My control is not as great as I want it to be yet, but you look at the positives," Ross said. "I've been able to pitch out of jams and put up zeroes."
Peavy wants the same kind of results posted by Ross, even if it was just his second start of the Cactus League on Monday afternoon at Camelback Ranch. But the right-hander remained upbeat and happy with his Spring Training work-in-progress, despite giving up six runs on eight hits over 2 1/3 innings.
He threw almost exclusively fastballs in his first home start against the Brewers but was able to work in more offspeed pitches in this trip to the mound.
"I threw a lot of breaking balls, some not so good," said Peavy, who walked two and struck out one. "I just had a hard time commanding the outer half of the plate, the glove side of the plate, getting the ball across.
"Obviously, I have a lot to work on as far as getting on time tempo-wise, getting your foot down and getting the ball away," Peavy said.
Oakland scored one in the first inning on Coco Crisp's single but took control with that five-run third. Collin Cowgill tripled home three off Leyson Septimo, who replaced Peavy with the bases loaded and one out when Peavy hit his pitch limit.
The White Sox rallied with four runs in the fifth off reliever Neil Wagner, with Alexei Ramirez, Kosuke Fukudome and Alex Rios all picking up RBIs.
Kurt Suzuki and Cowgill had three hits apiece for the A's, while Alejandro De Aza, Rios and Ramirez had two hits each for the White Sox.
Up next for A's:
Yoenis Cespedes will be back in the lineup Tuesday when the A's host the Brewers at Phoenix Municipal Stadium for a 1:05 p.m. PT start. The Cuban outfielder, who homered in his first game on Saturday, is expected to play in center and will be joined in the lineup by designated hitter Manny Ramirez. On the mound, right-hander Jarrod Parker continues his bid for a rotation spot. The game will be broadcast via a free audio webcast on athletics.com.