The Braves jumped out to an early 4-0 lead but the White Sox came storming back in the third inning.
That brought up Rios, who pounded a pitch deep into the left field bleachers, putting the Sox ahead to stay.
Phegley finished 3-for-4 with an RBI and two runs scored. De Aza went 4-for-5.
Jake Peavy (7-4) overcame a rocky start to collect the victory, allowing four runs (two earned) on seven hits over six innings. He didn't issue a walk and struck out three as the White Sox evened their three-game interleague set.
Maholm (9-9) was pounded for seven runs on seven hits over just three full innings.