Joakim Soria was told before the game he would be Texas' closer. He gave up his first run of spring training in the eighth.
Escobar hit a bases-clearing double in the Royals' five-run second inning. Escobar also doubled in the fifth and scored on Justin Maxwell's homer.
"He was really good with the bat and glove today," Royals manager Ned Yost said.
Vargas scattered eight hits over seven innings, allowing one run, while walking one and striking out five.
"Overall, it was a good day," Vargas said. "Kept some innings from being prolonged. They are going to get on base. That's how the game is, but you can't dwell on it. The key is to make that pitch to get you out of the inning. I was able to get up close to 100 pitches without feeling really fatigued, so that's a good sign."
Tommy Hanson, who is bidding for a Rangers' rotation spot, gave up seven runs on seven hits and three walks in 5 2/3 innings.
"It's never a good time to go out there and have that performance," Hanson said. "Spring is coming to an end and obviously I wish I'd thrown better and have had more balls down. But I didn't."
Rangers rookie Michael Choice hit a three-run homer in the eighth and raised his average to .340.
Rangers: Hanson threw 96 pitches, 56 for strikes, in 5 2/3 innings. His ERA jumped to 6.43.
Royals: Vargas lowered his ERA to 2.14, throwing 63 strikes in 96 pitches.
"Vargas was really good," Yost said. "He got us through seven innings under 100 pitches. That's a perfect scenario right there. He gives up a hit and he gets back on the mound and makes pitches. He's real efficient with his pitches. His command is really, really good, as good as anybody we've got at painting those corners.
"I'm real pleased where he's at. He's ready to go."
SCHEPPERS WINS ROTATION SLOT
Tanner Scheppers, who has been used strictly as a reliever in the majors, will be in the Rangers' rotation to begin this season.
Scheppers, a 27-year-old right-hander, has made only eight starts, seven in 2010, in the minors. He threw a career-high 76 2/3 innings last year.
"That would have to be a concern, simply because he's never done that before," Rangers manager Ron Washington said.
Alexi Ogando, who made 18 starts last year, will return to the bullpen. He is 7-1 with a 2.46 ERA and three saves in 108 career relief appearances.
FELIZ TRYING TO WIN JOB
Neftali Feliz not only failed to win the closer's job, but the 2010 American League rookie of the year has yet to earn a bullpen spot.
Feliz, who logged 72 saves in 2010-11, had reconstructive elbow surgery in August 2012 and has yet to return to his previous form.
"We told him he wouldn't be the closer," Washington said, "but he still has a chance to make the team."
Could Feliz open the season in the minors?
"I didn't say that," Washington said. "I said he's got to win a job on the team."
Royals left-hander Danny Duffy, attempting to earn a bullpen spot after failing to make the rotation, walked two and gave up Choice's three-run homer in his only inning.
Rangers: Elvis Andrus (right forearm soreness) is still not healthy enough to make the throws as a shortstop, but was the DH on Saturday.
Royals: Omar Infante (right elbow bone spur) remains sidelined. "I think Omar's going to be ready," manager Ned Yost said. "If he has another hiccup, he won't be. But if everything is going to plan, he'll be ready for opening day."
RANGERS ROSTER MOVE
The Rangers assigned infielder Brent Lillibridge to minor league camp. Lillibridge appeared in nine games with the Cubs and 11 with the Yankees last year.