Rodon, the third overall pick in the 2014 Draft, walked one and made good use of his fastball and slider in a start that covered 31 pitches, including 21 strikes. The only hit he allowed was a first-inning single to Kemp.
The Padres sent two candidates for the No. 5 spot in their rotation to the mound -- Odrisamer Despaigne and Brandon Morrow. Both showed well.
Despaigne allowed two hits over two innings with one walk and two strikeouts.
Morrow, who signed a one-year deal with San Diego in December, also had two innings of two-hit ball, striking out one with one walk.
"We are always evaluating our pitching ... the 29 pitchers in camp, how they're going to fit," Padres manager Bud Black said. "This is just another day along that path of trying to figure out who is going to make our team."
Kevin Quackenbush and Nick Vincent, important pieces to the Padres' bullpen a year ago, each tossed an inning. Vincent struck out two in the top of the sixth inning.
White Sox right fielder Avisail Garcia had hits in his first two at-bats and added a stolen base.
San Diego catcher Tim Federowicz gave the Padres a 1-0 lead with an RBI double in the fifth inning that allowed pinch-runner Abraham Almonte to score from first base. Almonte was running for Clint Barmes, who reached on a single.
Kemp then followed with his second hit of the game, which drove in Ramiro Pena, who was running for Federowicz.
Up next: The Padres play their first true road game of Spring Training -- away from the Peoria Sports Complex, that is -- when they face the Giants at Scottsdale Stadium on Saturday (12:05 p.m. PT, MLB.TV). James Shields, who last month agreed to the largest contract (four years, $75 million) in club history, makes his first appearance and is expected to go two innings. Oddly enough, the last time he took the mound was for Game 5 of the World Series -- against the team and pitcher (Madison Bumgarner) he'll face Saturday. The game will be broadcast on radio AM 1090.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.