Bergman allowed two hits and struck out one while finishing the game. With left-hander Jorge De La Rosa possibly headed for the disabled list with a groin injury, the 26-year-old Bergman might be an option for Colorado's rotation.
"He had real good command, that's what he does," Rockies manager Walt Weiss. "Both sides of the plate. He's got a great feel" for pitching.
Jose Abreu connected for his first spring homer, but White Sox left fielder Melky Cabrera committed two errors in Colorado's six-run fifth inning.
Cabrera, who signed a $42 million, three-year contract with Chicago over the winter, misplayed Matt McBride's single, allowing the Rockies to tie it at 5. Then he dropped Rafael Ynoa's fly ball near the line for a two-base error as McBride scored.
Rockies starter Jordan Lyles hit three batters, walked two and was charged with five unearned runs in four innings.
Gold Glove second baseman DJ LeMahieu committed an error in Chicago's five-run fourth. But he also made an outstanding play to start a double play on Gordon Beckham's hard grounder up the middle in the second inning.
Beckham hit a two-run double and scored on Tyler Flowers' first spring homer in the fourth.
Rockies: Lyles was acquired in a December 2013 trade with Houston and went 7-4 with a 4.33 ERA in 22 starts with Colorado last year. The right-hander is a sure bet for the Rockies' rotation, but he was all over the place against Chicago.
Lyles said he feels he is a long way ahead of where he was a year ago, and Weiss agreed, especially on the development of the changeup.
"It's a weapon now," Weiss said. "Last year it was an occasional changeup and he was more of a sinker-slider guy, and it was tougher for him to finish at-bats. Now he can get a swing-and-miss or bad contact with it."
White Sox: John Danks retired 11 of his first 12 batters, with only an error by third baseman Emilio Bonifacio spoiling the streak. But starting with Wilin Rosario's single in the fourth, the veteran lefty gave up four hits and four runs before leaving in the middle of Colorado's six-run fifth.
Danks said his changeup is getting better each time out. He is scheduled for two more starts before the regular season begins.
"It feels good to string together a few innings and feel good about what I did," Danks said. "As long as I get weak contact and throwing the ball over the plate I feel better about that than the strikeouts."
Before the game, White Sox manager Robin Ventura said he wasn't concerned that Abreu was still looking for his first homer of the spring.
"It would be nice, but he's so focused on his approach right now," Ventura said. "I think when you leave here you want to be able to have that, and the approach part for him trumps trying to go out there and hit homers. He can, and he's done it before."
And sure enough, Abreu then hit a drive to center off Jairo Diaz in the fifth inning. He is batting .459 (17 for 37) this spring.
The White Sox optioned infielders Leury Garcia and Andy Wilkins and pitchers Onelki Garcia and Frankie Montas to the minors. Montas will go to Double-A Birmingham; the others were headed for Triple-A Charlotte.
Rockies: RHP Eddie Butler, another contender for Colorado's rotation, faces the San Francisco Giants at Scottsdale Stadium on Wednesday.
White Sox: First-round pick Carlos Rodon starts against the Kansas City Royals in Surprise on Wednesday night. Brett Austin, a fourth-rounder who was Rodon's catcher at North Carolina State, will be behind the plate.