Aviles made his seventh consecutive start at shortstop and is batting .321 with three RBI in eight games for Kansas City, which is last in the American League in runs and could use some offensive punch.
Aviles was leading the Pacific Coast League with 37 extra-base hits and hitting .338 when he was promoted May 29.
"Mike's done well," Royals manager Trey Hillman said. "He had a short swing on that home run and got us back in the game. Hopefully, he keeps it going. It's another bat in the lineup."
"It all just went so quick," Aviles said. "It was the first pitch I saw. I was trying to get it in the gap to try to get a run. I got a good piece of the bat on it and drove it out of the park. I usually don't jog around the bases on a homer. It's just a habit out of respect to the pitchers to run quick around the bases."
Aviles spent five seasons in the minors before making his big league debut this year.
"The past week means a lot," he said. "It is always good when you feel like you're part of a team and that's how I feel right now."
The Rangers led 5-4 before Kansas City pushed across two runs in the eighth. Pinch-runner Tony Pena Jr. scampered home on Frank Francisco's wild pitch and Gordon scored on Jose Guillen's sacrifice fly to give the Royals the lead.
"It's good to see us come back after getting down," Hillman said.
Francisco (1-2) allowed two runs and two hits in one inning.
Texas scored four runs in the second to take a 4-2 lead. German Duran hit a three-run drive with two outs and Ian Kinsler followed with his ninth homer of the season. It was the fourth time this year the Rangers have hit back-to-back home runs.
Hurley went six innings for the Rangers in his major league debut. The 22-year-old right-hander allowed four runs and six hits, walked none and struck out three.
"I thought he was great," Rangers manager Ron Washington said. "I thought as the game progressed he relaxed a little bit more and he got better as the game progressed. He threw strikes and didn't walk anybody. He put the ball in play and that's what you want to see."
Gordon's two-run homer on a full-count pitch from Hurley gave Kansas City a 2-0 lead in the first.
"Both home runs were on four-seam fastballs," Hurley said. "When you're in the big leagues you expect nothing less than if you miss they're going to hit it. I wasn't going to give anybody any free passes today. I was going to throw three strikes to everybody if I had to, and I went out there and I did it."
Brian Bannister threw a career-high 127 pitches in seven innings for the Royals. He allowed four runs and six hits, walked none and struck out four.
"I wasn't pitching poorly, I just made two bad pitches and gave up two home runs and the rest of the day I pitched pretty good," Bannister said. "With that lineup, I was very happy with the outing."
- Guillen extended his hitting streak to seven games and is hitting .552 in that span.
- Murphy leads all American League rookies with 47 RBI.