Kinsler homered twice, a trio of relievers pitched shutout ball and the Rangers became the first visitors to win a series at Yankee Stadium since mid-June, beating New York 7-2 on Thursday.
"This sends a statement that we can play with anyone out there," Grilli said.
Davis also had a three-run homer for Texas, which took two of three from the team with the best record in the majors.
"We're treading into territory we've never been in before," Texas manager Ron Washington said.
Cut by the Colorado Rockies this season and activated from the disabled list over the weekend, Grilli (2-2) entered in the fourth with Texas ahead 3-2. He escaped a first-and-third jam by getting Alex Rodriguez on an easy comebacker and pitched 2 1/3 innings.
Grilli was aware he'd done well against Rodriguez -- the Yankees' cleanup man is 0-for-5 facing him. Grilli also realized his early relief appearance was an important one.
"It's not always so glamorous, but sometimes it's the tipping point," he said.
Neither starter let the other side hit the ball much, either -- for different reasons.
Through 7½ innings, in fact, only 33 of 66 hitters overall managed to put the ball in play. By the end, the teams combined for 25 strikeouts and 11 walks.
"Obviously, I had a good breaking ball today," Burnett said. "Everything was there."
Catcher Jose Molina went to the mound to talk with Burnett, and Kinsler lined the next pitch over the left-field wall.
"Just be ready for a strike. You can't give him anything," Kinsler said. "You can't sit on one pitch."
Kinsler said he "got lucky and ran into a heater." Burnett said it was a slider in the wrong spot.
Kinsler also connected for a solo shot in the eighth off David Robertson, giving him a career-high 28 home runs.
Burnett fell to 0-4 in six starts this month. He was tagged for a career-high nine runs by Boston in his last outing and shook off catcher Jorge Posada several times, leading to talk they weren't on the same page.
Posada missed this game, a day after a foul tip bruised his left ring finger. The Yankees thought the injury might sideline him for a few days.
"He's still day to day. Hopefully he'll be better tomorrow," manager Joe Girardi said.
Davis hit a three-run homer in the seventh off Phil Coke. He also made a nifty play at first base to end the sixth after Derek Jeter's hard grounder bounced off his glove -- Davis retrieved the ball and made a blind, backhanded flip to Grilli covering the bag.
"It was close your eyes and throw it," Davis said.
Texas became the first team to win a series at Yankee Stadium since Washington won two of three from June 16-18.
Mark Teixeira hit a pair of early RBI singles for the Yankees, both times driving in runners who drew walks from Nippert and stole a base.
Yankees pitchers struck out 15. ... Kinsler has five multihomer games this year and seven in his career. ... Texas has hit 190 homers, second in the majors to the Yankees' 195. ... Texas' Michael Young extended his hitting streak to 14 games, the longest active string in the majors. ... Tennis star Andy Roddick threw out the first ball. Known for his fastball serves, he went with a lob. A-Rod of racket fame waved at A-Rod of the Yankees before the ceremonial pitch.