Palka launched a 1-2 pitch from Neil Ramirez (0-2) six rows up into the left field seats for his 18th homer. The shot ended Chicago's three-game losing streak.
Jace Fry (1-2), the White Sox's third reliever, got the last two outs for his first major league victory.
Rodon and Cleveland rookie Shane Bieber tossed crisp scoreless ball deep into the game. Each was relieved after 106 pitches.
Rodon was sharp for a sixth straight start and has a 1.27 ERA in the span. The left-hander struck out five and walked two - both in the first and the only time Rodon allowed two baserunners in an inning - before being replaced by Xavier Cedeno to start the ninth.
Rodon allowed doubles with one out to Francisco Lindor in the third and Jose Ramirez in the sixth, but neither advanced.
Bieber matched Rodon through 6 2/3 innings, yielding only three hits - including two doubles - while striking out eight and walking two in this first career start against the White Sox. The right-hander was replaced with two outs in the seventh by Oliver Perez
The White Sox managed two baserunners only once against Bieber, in the second with two outs.
Yoan Moncada lined a double to the right-center wall to lead off the fifth, but didn't advance as Bieber struck out Ryan LaMarre and Nicky Delmonico before getting Yolmer Sanchez to fly out.
The White Sox used three relievers as they escaped a Cleveland threat in the ninth.
Cedeno walked Melky Cabrera to lead off, then Juan Minaya got Brandon Guyer to ground into a force at second. Yonder Alonso greeted Jace Fry with a single, but Fry struck out Jan Gomes and got Jason Kipnis to pop out.
AL Central Leading Cleveland is 35-17 against division opponents.
Indians: DH Edwin Encarnacion left the game in the third with left arm discomfort and was replaced by Cabrera. .. Manager Terry Francona had no update on OF Leonys Martin, who was placed on the 10-disabled list on Thursday with an unspecified illness that's not baseball-related. Francona, however, hinted Martin's condition could be serious. "What I would say, if you are into saying prayers and things like that, keep him in your thoughts," Francona said. "He's at the (Cleveland) Clinic and getting good care." ... RHP Josh Tomlin (strained right hamstring) and RHP Nick Goody (right elbow inflammation) threw bullpen sessions in Cleveland on Friday. Tomlin is expected to throw a rehab start at Triple-A Columbus on Monday and Goody will head to the Arizona League for a rehab assignment.
White Sox: Manager Rick Renteria said the team will know in the next couple of day if INF/OF Leury Garcia (left hamstring strain) will need a rehab stint. Garcia was injured on Sunday at Tampa Bay.
UP NEXT: Indians RHP Trevor Bauer (11-6, 2.25) takes the mound against White Sox RHP James Shields (4-13, 4.50) on Saturday night. Bauer has won his last three starts, and is 4-0 in his last seven with a 1.76 ERA.
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