Archer (10-8) struck out three and walked four in 6 1-3 innings. The right-hander is 2-0 with a 1.83 ERA in his last three starts.
"I think Archer was just so good we didn't get anything going," Ventura said. "He throws it hard, he's got a breaker."
McGee finished the three-hitter for his 18th save. The White Sox had two runners on with no outs, but McGee retired the next three batters.
Franklin connected against Hector Noesi in the second inning, belting a drive to right after a leadoff double by James Loney. Franklin had to wait 79 seconds for a replay review to determine if it was fair after it hit off an overhanging catwalk at Tropicana Field.
Franklin was acquired from Seattle in the three-team trade that sent David Price to Detroit on July 31. He was playing in his fourth game for the Rays, having been recalled from Triple-A Durham on Monday.
It was Franklin's first homer since Aug. 29, 2013, with Seattle.
Noesi (8-10) gave up three runs and six hits in six innings. He is winless in four September starts.
"I should be better than that," Noesi said. "I lost. I'm not happy."
Plate umpire Marvin Hudson took a hard foul by Chicago's Dayan Viciedo off the mask in the ninth, but stayed in the game after being checked out by a trainer. The trainer also looked at Hudson after the game.
White Sox: 1B Paul Konerko flew out as a pinch hitter in the ninth after going 0 for 4 Friday night in his first game since breaking his left hand Sept. 2. Konerko, who is retiring after the season, probably will be in the lineup Sunday, according to Ventura.
The White Sox have not homered in five straight games, their longest such stretch of the season.
RHP Nathan Karns (1-0) is scheduled to start the Rays' home finale Sunday against White Sox LHP John Danks (9-11). Karns allowed two hits over seven scoreless innings in his first start of the season Sept. 12 at Toronto, while Danks is 0-2 with a 5.73 ERA over his last four road starts.
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