Feliz and Soria are former AL All-Star closers. They both pitched sparingly last year while rebounding from elbow surgery.
Feliz surrendered three hits, including a two-run homer to Tyler Flowers in the sixth.
"It was a fastball out over the plate," Rangers manager Ron Washington said. "He tried to go down and away and left it up."
Soria worked a flawless fifth, throwing nine strikes in 16 pitches.
"Soria was Soria, a pro," Washington said. "He just went out there and got his three outs and got off the field. Pitching is about hitting your spots and Soria can hit his spots."
Adrian Beltre drove in two runs with two singles as the Rangers took a 4-1 lead. Beltre, Prince Fielder, Jurickson Profar and Brent Lillibridge combined to go 8 for 8 with four RBIs in the first two innings.
Adrian Nieto, a Rule 5 draft pick, and Jared Mitchell delivered two-run singles in Chicago's five-run ninth off three relievers.
White Sox: Felipe Paulino's Chicago debut was not pretty. The first four Rangers collected hits and nine of the 13 batters he faced reached base (eight hits and a walk) in 1 2-3 innings. He threw 47 pitches and committed a balk. Paulino had Tommy John surgery in 2012 and pitched just 27 2-3 innings last season in seven minor league rehab starts.
"The good thing now, my arm feels OK," Paulino said. "It is how I wake up tomorrow, but I believe I'll be OK. I feel confident about this. The past is past. I'm looking forward.
"They hit me. Find the holes, that's part of the game. I will try to build it up and hopefully next time will get better luck."
Rangers: Left-hander Robbie Ross, who was 10-2 with a 2.62 ERA in 123 relief appearances for Texas the past two years, yielded one run on two hits in two innings. He walked one and hit leadoff hitter Adam Eaton. Ross, who started in the minors, is a rotation candidate.
"Coming in, there's a chance instead of throwing 70 innings I might throw 200 innings," Ross said. "I'm thinking of getting my legs as strong as I possibly can, doing the things I need to do to get strong. We don't have Derek (Holland, knee surgery) here, so that opens up a spot. Obviously, I'm wanting to get a starting spot. That's what I'm battling for, but at the same time, I'm battling for a relief spot. Whichever is fine with me, it's just that I'd like to get that starting spot."
Rangers: Outfielder Shin-Soo Choo has left triceps soreness and did not play. He is expected to miss a few games to quiet it down. Shortstop Elvis Andrus missed his second straight game with right flexor tendinitis.
White Sox: Paul Konerko was scratched as a precaution after feeling sick.
DUNN FOR OSCAR
White Sox slugger Adam Dunn left to attend the Academy Awards in Los Angeles. Dunn had a cameo non-speaking appearance as bartender Neddie Jay in the Oscar-nominated movie "Dallas Buyers Club."
HAT IN RING
Tanner Scheppers, who worked exclusively as a reliever in his first two Texas seasons, is also a starting candidate. He tossed two scoreless innings, allowing one single, striking out one and hitting two batters.
"I still feel kind of rusty with everything," Scheppers said. "It was my first time to throw to J.P. (Arencibia, catcher), something I actually really enjoyed, but we've still got to learn each other's moves."