In the opener, Kip Wells (2-0) won his second big league start to give the White Sox a 4-2 victory and was then optioned back to the minors after the game.
Durham hit a two-run homer and a RBI single off Ryan Glenn (2-4) in the second game for the White Sox, who had lost a season-high seven straight before the sweep.
Texas has lost six of its last nine games and saw its lead over Oakland in the AL West slip to 5 ½ games.
After Magglio Ordonez's RBI groundout in the first gave Chicago the lead, Brook Fordyce hit a run-scoring double and scored on Durham's single in the second to make it 3-0. Chris Singleton added an RBI double in the third.
Durham hit his 13th homer in the fourth to make it 6-0 and Caruso added a run-scoring groundout in the sixth.
Glynn gave up six runs and nine hits in five-plus innings.
Rusty Greer hit solo homers in both games and had a two-run single in the seventh of the nightcap for the Rangers. Greer has 11 homers.
Lee Stevens added a solo homer in the ninth for Texas.
In the opener, Wells (2-0), Chicago's No. 1 pick in the 1998 amateur draft, gave up one run, three hits and three walks in five innings. He struck out six and overcame a 56-minute rain delay in the top of the third.
The 22-year-old right-hander won his major-league debut Aug. 2 against Detroit, allowing two runs one earned in 5 1-3 innings.
He was sent back to Double-A Birmingham following that start and recalled before Friday's doubleheader. Immediately after the first game, Wells was again optioned to Birmingham.
"I know it's part of the game," Wells said. "I didn't expect to walk in here and take over. They've got five guys in the rotation. Six won't do it."
Bob Howry pitched the ninth for his 19th save in 23 chances.
Texas cut the lead to one in the fourth, when Wells balked with runners on first and second and Lee Stevens followed with an RBI groundout.
Ordonez, who went 3-for-4 in the opener, hit an RBI single in the fifth to make it 3-1 and Caruso added a run-scoring groundout in the sixth.
Greer's 10th homer, off Keith Foulke in the eighth cut Chicago's lead to 4-2.
Burkett threw a season-high 109 pitches over 5 1-3 innings. He allowed four runs, 11 hits and two walks.
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