Avisail Garcia and Dayan Viciedo singled to put runners on the corners with two outs in the ninth, and Phegley hit a single to left to win it.
The Sox win also snapped the Rangers nine-game road winning streak.
Paul Konerko hit a two-out single to left in the fourth inning with Alexei Ramirez at second base, Rangers left fielder Craig Gentry threw out Ramirez at home. Ramirez looked to have scored but home-plate umpire Hunter Wendelstedt called him out as former White Sox A.J. Pierzynski’s leg blocked Ramirez from touching the plate.
In the sixth inning, Alex Rios hit a two-run home run to left field off Sox starter Hector Santiago to score Pierzynski and give the Rangers a 2-0 lead.
Adam Dunn hit a two-run home run, his 29th of the season, in bottom half of the sixth off Rangers starter Yu Darvish.
Santiago allowed five hits and two earned runs, struck out six and walked three in six and one-third innings.
Darvish went seven innings, gave up two runs and six hits, struck out 11 and walked three.
Sox left-hander John Danks (3-10, 4.22 ERA) and Rangers right-hander and former Cub Matt Garza (3-1, 4.32 ERA) are the projected starters for the series finale in Chicago on Sunday afternoon at 1:10 p.m. CT (CSN, WSCR 670).