Abreu singled to center in the first to score Adam Eaton, who had doubled to left off of Texas starter Joe Saunders. In the fourth, Abreu singled to left off of Tommy Hanson to bring home Marcus Semien. Abreu followed with a single to center in the seventh for hit No. 3.
It's Abreu's professional approach of taking what is given to him at the plate and not consistently swinging for the fences that has impressed the White Sox.
While the White Sox were connecting for four runs (three earned) on six hits over three innings against Saunders, John Danks was shutting down the Rangers. Danks yielded four hits over five scoreless innings, striking out one and walking three.
There was a little luck involved, as shown in the Rangers' second and fourth innings. Mitch Moreland was hit by a pitch to open the second but was doubled off of first on a perfect throw to first from Eaton when Moreland broke on Engel Beltre's line drive to center.
Moreland walked to open the fourth and once again was put in motion. Beltre hit another line drive, but this one found shortstop Alexei Ramirez and his throw to first doubled up Moreland.
"I'll take that every time out. It's true: Double plays are the pitchers best friend," Danks said. "There were a couple of times an inning could have got ugly on me and you see the play Eaton made and the last inning Alexei. I get a pickoff or whatever.
"Getting outs, I've never gotten a bad out. It's nice to know you got a defense behind you. You can stay aggressive and let these guys work for me."
Look no further than broken bats in at-bats for Prince Fielder, Adrian Beltre and Leonys Martin for proof that Danks' cutter was once again sharp in this Cactus League contest.
Robinson Chirinos singled twice and walked for Texas, driving in the Rangers' first run in the seventh off of reliever Scott Downs. Gordon Beckham homered for the White Sox, also coming off of Saunders, who struck out four.
"I made a couple of mistakes that hurt me but my arm felt good and the ball was coming out of my hand good," Saunders said. "My command was a little shaky at times but that will get better with more bullpens."
Trailing 5-1 in the eighth, Texas scored five runs off of White Sox relievers Mitchell Boggs and Donnie Veal. Jim Adduci and Bryan Petersen doubled home runs against Boggs, while run-scoring singles from Rougned Odor and Michael Choice put the Rangers ahead.
That lead held up until Matt Davidson's two-out solo homer in the eighth evened the score at 6-6.
Up next: Alexi Ogando, the No. 2 starter in the Texas rotation, will be on the mound Wednesday when the Rangers face the Angels in Surprise at 3:05 p.m. CT. Ogando is 1-1 with a 3.60 ERA in two starts so far this spring. He will be opposed by Joe Blanton of the Angels and the game can be heard on the texasrangers.com webcast with Steve Busby calling the action.