Jason Coats hit a go-ahead home run in the eighth off Dodgers reliever Jose De Leon, before Rob Segedin sent one out to lead off the ninth against White Sox reliever Jake Petricka.
Davidson's two-run homer in the seventh off De Leon snapped a 3-3 deadlock, but Austin Barnes clubbed a two-run shot, his fourth of the spring, off White Sox reliever Dan Jennings in the eighth to forge another tie.
Andre Ethier hit a two-run shot to the opposite field in the second inning, and Kiké Hernandez followed with a solo homer in the third as the Dodgers got off to a strong start against Jose Quintana, who allowed three earned runs over four innings.
"I threw more pitches after four innings, I went to the bullpen and threw 20 more," Quintana said. "Next time, I'll throw better and throw more pitches. … I was working on the corners, especially back door and in. I made a couple of mistakes, but I'll work on it and try to get better."
Kenta Maeda made the start for the Dodgers, allowing two unearned runs on four hits in 3 2/3 innings in his second outing of the spring. The runs scored in the fourth inning because Brett Lawrie reached and took second on an error by Dodgers first baseman Adrian Gonzalez -- one of four errors on the day by the Dodgers.
Lawrie scored the first White Sox run on a single to center by J.B. Schuck, and Alex Avila came around to score when Adam Eaton delivered a soft fly ball to right field. The White Sox tied the game in the sixth when Austin Jackson scored as Schuck reached on an error.
Up next for Dodgers: Fifth-starter candidate Zach Lee gets another start when the Dodgers host the Rockies on Wednesday at 1:05 p.m. PT. Nineteen-year-old Julio Urias is scheduled to pitch second, and Justin Turner is expected to return to the lineup after a day off following his Cactus League debut on Monday.
Up next for White Sox:Carlos Rodon looks to continue a very strong start to the spring in his third Cactus League outing when he meets the Brewers at Maryvale Baseball Park on Wednesday at 3:05 p.m. CT, live on MLB.TV and Gameday Audio. Rodon has gone 2-0 with a 0.00 ERA in his first two outings, covering seven innings.
John Schlegel is a national reporter for MLB.com. You can follow hium on Twitter @JohnSchlegelMLB.