Texas had its four-game winning streak snapped and wasted another strong outing by Yu Darvish, who gave up two runs in seven innings while striking out 11.
Avisail Garcia singled with one out and went to third on Dayan Viciedo's hit one out later. Phegley then hit a line shot to left off Tanner Scheppers (5-2) to end the game.
Nate Jones (4-4) pitched an inning for the victory.
Chicago trailed 2-0 before Adam Dunn hit a two-run homer in the sixth inning to tie it. Alex Rios had hit a two-run shot in the top half of the inning for the lead.
NOTES: As part of the Civil Rights Game proceedings, Aretha Franklin and Bo Jackson were honored during the pregame ceremonies. Franklin couldn't attend and former Tigers great Willie Horton accepted in her honor. ... Before the game, White Sox manager Robin Ventura was asked how good Jackson might have been without the hip injury. "It's hard to describe him to people now of how strong, fast and smart of a baseball player he was," Ventura said. "He was good regardless. We'll never know, but I haven't seen anyone that's had his abilities since." Rangers RHP Nick Tepesch threw live batting practice on Saturday and reported no issues. ... RHP Matt Garza (9-2), a former Cub, makes his return to Chicago Sunday when he faces John Danks (3-10) in series finale.