Abreu homered in the first inning, doubled in the second, singled in the seventh and hit a triple in the bottom of the eighth inning.
With a chance at the cycle, Abreu momentarily lost his balance during his eighth-inning dash to third base.
"My legs weren't responding, but I'm a warrior,'' Abreu said.
The 30-year-old Cuban joined Jose Valentin (2000), Chris Singleton (1999), Carlton Fisk (1984), Jack Brohamer (1977) and Ray Schalk (1922) as White Sox players who have hit for the cycle.
"Initially, off the bat, I knew he wasn't going to stop,'' White Sox manager Rick Renteria said. "I knew he was going to give an effort. Those things don't happen very often."
Abreu hit a foul ball off his left shin earlier in the at-bat. He has a .363 batting average and 10 homers in his past 25 games.
"No, no,'' Abreu said when asked whether he would agreed to take a seat.
"I don't like to leave the games,'' Abreu said. "I like to play the whole game.''
Giants starter Jeff Samardzija, who gave up six runs and eight hits in 5⅔ innings, praised his former teammate's hitting and work ethic.
"He's always been an amazing hitter,'' said Samardzija, who spent one season with Abreu in Chicago. "I got to see it firsthand. Dude's the first guy in the cage every day, first guy at the park.''
The seven cycles this season are the most in a season since 2009, when there were eight.
The Associated Press contributed to this report.