Already missing No. 1 starter Chien-Ming Wang and right-hander Phil Hughes from a top-heavy rotation, the Yankees received more bad news when Chamberlain exited with a stiff right shoulder in the fifth.
He'll return to New York on Tuesday for tests, and manager Joe Girardi said the 22-year-old right-hander probably will miss a turn in the rotation. Anything more than that would be a major blow to the injury-depleted Yankees, who are 2½ games behind Boston for the AL wild-card lead.
"He has a little stiffness. We believe it's muscular," Girardi said. "We'll see what the tests reveal. I don't think it's serious. I don't think he'll be shut down a long time."
But the Yankees are likely to be cautious with the hard-throwing Chamberlain -- even in a tight playoff race. He said he's never had a similar problem in his shoulder.
"My strength was fine. My velocity was fine," Chamberlain said. "We'll have tests just so we can rule out everything. It kind of grabbed and it was a little bit tighter in the fifth inning."
After the game, Girardi also revealed that star closer Mariano Rivera was unavailable for the second consecutive game because of back spasms.
"We were a little short-handed," the manager said. "He said he felt OK, but I've got to hear that he feels great."
Girardi said he expects Rivera will be ready to pitch Tuesday night.
Leading off the bottom of the ninth on a 101-degree night, pinch-hitter Milton Bradley drew a walk from Damaso Marte (0-1). Travis Metcalf ran for Bradley and, one out later, Marte walked Gerald Laird.
After Michael Young flied out, Josh Hamilton walked to load the bases and Byrd hit the next pitch from Marte into the right-center seats. Byrd was mobbed at the plate by his teammates after circling the bases with his seventh homer.
"I knew it was gone right when it came off the bat. He threw a fastball and I put a good swing on it," Byrd said after his second grand slam this season and the fifth of his career.
Young and David Murphy also homered for the Rangers, who earned their first home win over the Yankees since July 19, 2006.
Texas moved five games over .500 (59-54) for the first time this season.
"It was big because of the way that we did it, against the team we did it against, factoring in the circumstances," Murphy said. "We'll ride this for a while."
New York threatened in the ninth when Johnny Damon hit a two-out double off Eddie Guardado (3-2). Derek Jeter hit a line drive to center, where Hamilton made a tumbling catch to keep the score tied at 5.
"It was a hard one," Girardi said.
Young's three-run shot off Chamberlain in the fifth gave Texas a 5-4 lead.
Byrd hit a two-out single in the fifth and when Chamberlain returned to the mound, assistant trainer Steve Donohue and Girardi ran onto the field to check on the former reliever, who pointed at his right shoulder.
Chamberlain, who gave up five runs and eight hits over 4 2/3 innings in his 12th major league start, walked to the dugout and was replaced by Edwar Ramirez.
Giambi's RBI single in the first gave New York the lead. Giambi led off the fourth with his 21st homer for a 2-0 advantage and, one out later, Cano went deep for the 10th time to make it 3-0.
Murphy's two-run shot off Chamberlain in the fourth -- his 15th -- pulled Texas to 3-2.
Rangers starter Vicente Padilla gave up four runs and six hits in six innings. Texas manager Ron Washington was ejected for protesting two balk calls in the fifth against Padilla.
Girardi started rookie Justin Christian in CF for Melky Cabrera, who came in for defense in the ninth. Girardi said Cabrera -- hitless in his last 12 at-bats -- could use a few days off. ... Chamberlain gave up three runs or fewer in each of his first 11 big league starts. ... Bradley missed his sixth straight start due to a strained left quadriceps.
Fantasy baseball daily notes: Pitcher and hitter rankings for Saturday
Prepare your fantasy baseball lineups confidently with our daily preview, including pitcher projections, hitter matchup ratings and more.
Fantasy Baseball Forecaster for Week 24: Sept. 20 to 26
Set your lineups for the week ahead in fantasy baseball with our forecaster preview, which features pitcher rankings, two-start pitchers, hitter ratings, schedule observations and more.
Fantasy baseball forecaster: Team hitting ratings
Stay ahead of the game and optimize your fantasy baseball hitters with our forecaster, which provides a rolling 10-day outlook of every team's hitting matchups as well as the potential for stolen bases.
Fantasy baseball forecaster: Pitcher projections for the next 10 days
Keep up to date on who is pitching -- and how well they're expected to do -- with our forecaster, which provides a rolling 10-day outlook of every team's probable starting pitchers with game score projections.
Either the Giants or Dodgers will face a wild-card elimination game? Here's why that's a good thing
This year, the Giants or Dodgers will see a 100-plus-win season come down to a winner-take-all wild-card game. That's something worth celebrating!
2021 MLB playoffs: Who is in, who can clinch next and full playoff schedule through World Series
October is right around the corner. Here's the latest on which teams will be playing in the postseason.