Pinch-hitter Rougned Odor doubled home Engel Beltre, who had singled, with the only run off Vargas in the seventh. Beltre and J.P. Arencibia each had a double and a single for the Rangers.
Vargas is the first Royals pitcher to go seven innings. He gave up eight hits with one walk and five strikeouts. He lowered his ERA to 2.14.
Right-hander Tommy Hanson, one of four candidates for the two remaining spots in the Rangers' rotation, was clipped for seven runs in his 5 2/3 innings.
"It's never a great time to go out and have that kind of performance," Hanson said. "The spring is coming to an end. I wish I pitched better. I wish I got my pitches down. I've just got to keep working."
In the second inning, the Royals loaded the bases on two walks and Danny Valencia's single with two outs. Norichika Aoki beat out a ground ball to shortstop Adam Rosales for the first run. Escobar followed with a long fly ball that eluded center fielder Leonys Martin and went for a three-run double. Eric Hosmer singled to right, scoring Escobar.
That gave the Royals a 5-0 lead, but Hanson ended the inning by picking off Hosmer at first.
Helped by Martin's diving catch, Hanson retired the next seven batters. But in the fifth Escobar got his third hit, a double to left-center, and Maxwell hammered a two-run homer down the left-field line. It was Maxwell's second homer.
"Some balls were up, some balls that were hit hard. I wasn't as consistent as I need to be down in the zone. The walks in the second inning didn't help," said Hanson, who is competing with Robbie Ross, Joe Saunders and Colby Lewis for the two rotation spots.
Ex-Royal Joakim Soria, who has won the job as Rangers closer, gave up an eighth-inning run on Johnny Giavotella's single and Ramon Hernandez's double.
The Royals (11-11-2) won the series against their complex co-tenants, three games to one. The Rangers are 7-14-3.
Up next: Left-hander Joe Saunders pitches against the Padres on Sunday in Surprise. Saunders is trying to win a spot in the Rangers rotation and a good outing on Sunday might do just that. Pedro Figueroa, Shawn Tolleson, Michael Kirkman and Neftali Feliz are also scheduled to pitch.