Ian Kinsler hit a leadoff homer, and Michael Young had three hits for the Rangers, who placed outfielder Josh Hamilton on the 15-day disabled list before the game due to a strained left rib cage muscle.
Prized Rangers prospect Derek Holland (0-1) relieved Scott Feldman with the bases loaded in the sixth. After whiffing on Holland's 94-mph fastball, Thome hit the next pitch off the top of the 14-foot wall in left to tie it at 3.
Getz, who had missed five games with a finger injury, tripled to put Chicago on top for good.
Bobby Jenks pitched the ninth and is perfect in six save chances.
Feldman, making his second start of the season, allowed three runs and three hits in 5 2-3 innings. He retired the first 11 before giving up a single to Carlos Quentin.
The right-hander allowed one hit through five innings and recorded the first two outs of the sixth before running into trouble.
Getz and Josh Fields singled, and Quentin walked to load the bases and chase Feldman from the game.
Thome, who missed the previous three games due to a sore left heel, has 16 RBIs in his last 10 games at Texas and 1,501 overall in his career.
The White Sox also used a two-out rally to take the lead in the seventh.
Scott Podsednik, who had his contract purchased from Triple-A before the game, had an infield single with two outs. Getz lined a pitch into right-center off Holland to give Chicago a 4-3 advantage.
