The defeat came after Alex Rodriguez had tied the game with a homer -- his 601st -- leading off the eighth inning to help send the game to extras. The homer took A.J. Burnett off the hook in an otherwise solid start in a contest in which Dave Robertson also pitched out of trouble.
Rivera started the 10th inning and Michael Young reached on an infield hit that Derek Jeter couldn't fire across the infield quickly enough, and Josh Hamilton dribbled a weak ground ball through the right side to put Rivera in trouble.
"They're hits. You can't do [anything]," Rivera said. "I tried to make good pitches, and it didn't work tonight."
A-Rod did his best to help the cause, making a full-body diving stop to his left and a strong throw to rob Vladimir Guerrero, but Murphy followed an intentional walk to Nelson Cruz with the game-winning knock that ended a steamy meeting between these potential playoff rivals.
"Hey, we beat him," Rangers manager Ron Washington said. "You don't get that opportunity very often, but tonight we did. It's a big one. It's a huge win, one that we needed. They're the best and we came with our best foot forward and got a win."
"It's tough. There's not too many guys that hit balls hard off Mo," Jeter said. "You tip your hat to them because they had some good at-bats. They were able to get that big hit there at the end."
In the eighth, Rodriguez crushed a 2-1 fastball from Frank Francisco onto the berm beyond the center-field fence, his 18th home run of the season, erasing an advantage that Texas had chiseled when Murphy turned around a first-pitch Burnett fastball for a two-run homer in the sixth.
It provided a scarce blemish on an otherwise encouraging evening as Burnett appeared to come back healthy from back spasms that delayed his scheduled start, which should have been against Boston on Saturday in New York.
"I wasn't trying to throw it down the middle," Burnett said. "It was supposed to be away. That was the only pitch I'll take back from tonight."
Burnett was also touched in the fourth inning, hanging a breaking ball to Cruz, who hammered it into the left-center field gap for an RBI double. Burnett completed seven innings, allowing three runs on six hits with two walks and four strikeouts.
"I felt fine out there," Burnett said. "I didn't feel anything. I wasn't overcautious. I knew with the heat and the back thing, I wanted to stay within myself. I haven't pitched in a long, long time, and I had to pitch."
The Yankees took a lead against Texas starter C.J. Wilson in the sixth, as Austin Kearns notched a one-out double and Lance Berkman walked to chase Wilson. Francisco Cervelli laced Darren O'Day's second pitch into center field for a run-scoring single.
New York had broken through against Wilson in the fifth, after a two-out walk to Jeter and a gift base with a throwing error.
Nick Swisher took advantage by belting a double off the wall in center field to tie the game, though Swisher ended the inning when he was gunned down at the plate by a considerable margin on a single to left.
As on Monday, when the Yankees went 0-for-9 with runners in scoring position in a 2-1 loss to Boston, the big hit seemed to elude the Bombers' bats.
"We had our opportunities to win this game," manager Joe Girardi said, "and didn't get it done."
Playing with a weakened lineup -- Mark Teixeira was excused to attend to the birth of his son and Robinson Cano was sidelined until the sixth inning with a fever -- the Yankees were 3-for-11 with runners in scoring position and left nine on base.
Their last promising opportunity came in the eighth, after Rodriguez's homer. A wild pitch and flyout moved pinch-runner Curtis Granderson to third base, but Cervelli popped weakly to first base and Brett Gardner grounded out against Francisco to end the threat.
"I think you go through periods like that at times, and other times pitchers bear down," Jeter said. "It's not like they're going to say, 'Here you go,' and let the run come in. They pitched well. It's the reason they've been playing well all year."