Murphy put Texas ahead for good with a drive just over the wall in right field in the sixth inning, and the Rangers spoiled the second three-homer game of Cabrera's career with an 11-8 victory on Sunday night.
"Very nice to be on the winning side," Murphy said. "Big game, fun game, national TV. We got to witness the best hitter in the game hit three homers."
Cabrera hit a three-run homer in the third and a solo shot in the fifth off Derek Holland, then went deep again in the eighth against Tanner Scheppers.
Cabrera, the 2012 AL Triple Crown winner and this year's leader in average and RBIs, went 4 for 4 with 5 RBIs and has 11 homers, one off the AL lead.
Still, the Tigers lost three of four to the Rangers, who won a four-game series for the first time after twice losing with a chance to do it.
"I've never seen a guy that puts the bat head on the ball as regular as he does, and on all pitches," Rangers manager Ron Washington said. "He's just a tremendous hitter. I'm very happy that he's headed to the airport."
Murphy's homer gave the Rangers an 8-7 lead and answered a bases-loaded double from Prince Fielder that put Detroit up by two after the Rangers had walked Cabrera intentionally even though first base wasn't open.
The go-ahead shot came against Jose Ortega (0-1), who hadn't allowed a run in six relief appearances covering seven innings.
"He got me down 0-2, and I was basically in battle mode to do whatever I could to prolong the at-bat and get anything done in that situation," Murphy said.
Robbie Ross (2-0) pitched a scoreless seventh inning for the win, and Joe Nathan worked around a leadoff walk in the ninth for his 13th save. He has saved all 36 career chances against Detroit.
The Tigers led 4-1 after Cabrera hit a three-run homer over the Texas bullpen in right-centre field in the third and added a solo shot to centre off Holland, who had given up just two homers in 57-plus innings this year after setting a club record by allowing 32 homers last year.
The Rangers had wiped out that lead — and another one after Fielder's double — and were ahead 11-7 when Cabrera hit another solo homer to centre off Scheppers.
"You don't feel very good when guys hit three home runs and you lose the game," Detroit manager Jim Leyland said. "Miggy never ceases to amaze you. I also take my hat off to Prince Fielder for that situation against a left-handed pitcher. Between the two of them, that was impressive."
The Rangers overcame Detroit's early lead with a four-run fifth highlighted by a pop fly double from Adrian Beltre that scored two and gave Texas a 5-4 lead.
Beltre went 4 for 5 with two doubles — both of them bloopers — and Geovany Soto and Lance Berkman had three hits apiece for the Rangers, who matched their season high with 18 hits.
Soto, who came in hitting .185, went 3 for 5 for the Rangers and had a fourth hit taken away when Andy Dirks caught a long drive to centre while crashing into the wall.
Dirks' catch on Soto ended a string of six straight Texas batters reaching in the sixth against Ortega (0-1), who had allowed only five base runners this season.
Soto started both Texas rallies with leadoff hits in the fifth and sixth innings, when the Rangers scored eight of their runs.
The light-hitting catcher had a double to start the fifth and scored on a groundout by Elvis Andrus. Murphy, who also had a sacrifice fly, walked and Berkman drove in Craig Gentry with a single before Beltre's second double, which slipped out of Detroit shortstop Jhonny Peralta's glove just inside the left-field line.
Washington scrambled to protect a 5-4 lead in the sixth, finally ordering reliever Jason Frasor to walk Cabrera with runners at first and second when he got behind 3-1. Washington brought in left-hander Michael Kirkman to face Fielder, but the lefty slugger ripped his second double to the wall in centre field.
Soto started the sixth with a single, and Andrus walked with one out before Murphy pulled a pitch from Ortega to right to put Texas ahead for good.
"Neither team quit fighting," Washington said. "It was a very good ballgame. It didn't matter what you tried to do. Each team came back, and we got the final blow and were able to sustain."
Holland and Detroit starter Doug Fister both went just 4 2-3 innings, the shortest outing of the year for Holland. The left-hander gave up eight hits and four runs with three walks and six strikeouts. Fister allowed nine hits and five runs.
Notes: Cabrera hit three homers on May 28, 2010, at home against Oakland. ... Rangers 2B Ian Kinsler went on the 15-day disabled list with bruised ribs, and top prospect Jurickson Profar was brought up from Triple-A Round Rock. Leury Garcia made his third straight start in Kinsler's place at second. ... Rangers C A.J. Pierzynski caught five innings in the first game of a rehab assignment at Double-A Frisco on Sunday. He went 0 for 3 with a strikeout and two groundouts. Pierzynski will catch RHP Colby Lewis' rehab start with Frisco on Monday.