McAllister gave up one hit and one walk. He has yet to allowed a run in four innings of Cactus League work.
"He looks like he worked really hard this winter," Indians manager Terry Francona said. "When you can pitch at a nice, comfortable velocity and you got a little extra, kind of shows how hard he worked. His arm strength is tremendous."
McAllister has already pinpointed something to improve.
"I might like to get a little bit more strikes on the inside to righties," he said. "But I'm able to drive the ball in there, which is something I'm working on and felt good doing it."
Jose Abreu, the top prospect in the White Sox organization, hit a two-run double and walked. It was his first hit of the spring through three games.
Joe Wendle singled home the winning run in the ninth.
Erik Johnson pitched three innings in his spring debut for the White Sox. He gave up one run on three hits with a walk and two strikeouts.
White Sox: The 24-year-old Johnson is among the top prospects for the White Sox.
"Today was just to get my work in and use all my pitches and get out of there healthy," he said. "I thought I could have done a little bit more throwing first-pitch strikes and working ahead, but overall it's a good one to start off."
White Sox: Pitcher Matt Lindstrom was scheduled to throw a bullpen session on Tuesday as he comes back from a left oblique strain. ... Scratched from Monday's game because of tightness in his right leg, Gordon Beckham returned to the lineup Tuesday, getting two hits and scoring a run.
Indians: Infielder Jose Ramirez (left thumb) was in the B game starting lineup on Tuesday. He singled in his first at-bat. . First baseman/outfielder Bryan LaHair (left wrist) is taking batting practice in the batting cage. ... Pitcher Matt Capps (rotator cuff) is still 7 to 10 days from beginning a return-to-throwing program. ... Pitcher Frank Herrmann (right elbow) was scheduled to throw a bullpen session on Tuesday with a simulated game on March 7. ... Pitcher Shaun Marcum (thoracic outlet surgery) is slated to throw bullpen sessions on March 5 and March 8.
BACK FROM HOLLYWOOD
Adam Dunn returned to the White Sox lineup after his trip to the Academy Awards for his role in "Dallas Buyers Club." He was hitless in three at-bats.
NOT BAD FOR A CATCHER
Position-shifting Carlos Santana got the start at third base for the Indians. On Sunday against the Mariners, he threw a ball wide of first base for an error.
But on Tuesday, he easily fielded a high chopper from Tyler Flowers for the out.
THE ANNIVERSARY OF MJ
Tuesday marked the 20th anniversary of the first time Michael Jordan put on a White Sox uniform in a game.
"It was fun. You're talking about a guy who was at the height of his career in another sport," said White Sox manager Robin Ventura, who was on that 1994 team. "It was crazy crowd-wise, you just didn't get (spring training) crowds like that back then."