The rookie accepted the challenge and flourished.
Allard (2-0) allowed six hits, struck out eight and walked none in his fourth start since he was acquired in a trade with Atlanta last month.
"These are obviously major league hitters. The more weapons you got in your back pocket to go at them and the more ways you can get people out, the longer you're going to hang around this game and the more success you're going to have," Allard said.
Calhoun and Nick Solak had two hits apiece for the Rangers, who had dropped seven of 10.
Jose Abreu had two of Chicago's eight hits, giving him 1,001 for his career. The White Sox had won three in a row.
Calhoun hit a two-run drive off Ivan Nova (9-10) in the sixth, giving the Rangers a 2-0 lead. It was his fourth homer in his last six games and No. 15 on the season, extending his career high.
Nova was charged with an earned run and six hits in 5 2/3 innings. He was 5-0 with a 0.85 ERA in his previous six starts.
"I'm just throwing better pitches," Nova said. "We keep working hard. We're still hungry and want to keep pitching better."
Santana added another two-run shot against Jimmy Cordero in the seventh.
In his previous start, Allard allowed a career-high seven runs in five innings against the Angels.
He found more success with his cutter Saturday.
"Came out a little flat in the first against Anaheim and obviously it showed," Allard said. "Came out with a little bit more of an emphasis on coming out of the gate early and coming out of the gate hot today and we did a good job executing from pitch one."
Heineman caught Anderson's drive to the center field wall to end the Chicago fifth with a runner on second.
JOINING THE CLUB
Abreu called reaching 1,000 hits "a really big accomplishment for me, my career, for my family and for this organization."
FRONT OFFICE CHANGES
Rangers director of player development Matt Blood is moving to the new role of director of baseball innovation. Assistant GM Mike Daly takes over as farm director.
Rangers: RHP Jesse Chavez (right elbow impingement) will possibly have surgery to remove a bone spur. "He's going to see doctor when we get back home and I think the possibility of getting that spur cleaned out is on the table," general manager Jon Daniels said. "If that were to happen, he would not come back this year. He should be 100 percent for next year." Chavez has been on the IL since Aug. 13. ... OF Hunter Pence (lower back tightness) was out of the lineup and not expected to play Sunday. He said the injury occurred Friday in the batting cage. ... OF Joey Gallo (right wrist surgery) will have to play through pain if he returns this season, manager Chris Woodward said. "It's a tolerance thing. He can't hurt it anymore," Woodward said. Gallo has been on the IL since July 24. ... OF Nomar Mazara (left oblique strain) is taking swings as he nears a return from the IL.
White Sox: OF Eloy Jimenez (right hip flexor soreness) returned as the designated hitter after he missed two games. The team is easing him back onto the field.
Rangers left-hander Brock Burke (0-0, 0.00 ERA) faces White Sox right-hander Reynaldo Lopez (7-11, 5.25 ERA) on Sunday. Burke will make his second career start after throwing six scoreless innings Tuesday against the Angels. Lopez is 3-3 with a 3.10 ERA in his last eight starts.
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