With the score tied 2-2 in the ninth, Garcia collected a one-out single off Tanner Scheppers (5-2).
After Conor Gillaspie fouled out, Dayan Viciedo singled, sending Garcia to third, and Phegley followed by lining a solid hit to left field to end the game.
Adam Dunn bashed a two-run homer for the White Sox, who won for the seventh time in eight games and evened the three-game series.
Nate Jones (4-4) was credited with the decision after tossing a scoreless top of the ninth.
Former White Sox outfielder Alex Rios connected on a two-run shot for American League West-leading Texas, whose four-game winning streak was snapped.