Murphy, who pulled a right oblique muscle on Aug. 9, lined a 1-2 pitch to center off Ronald Belisario to score pinch-runner Chris Dickerson and give the Indians their 11th walkoff win of the season.
Yan Gomes started the inning with a triple to left off Maikel Cleto (0-1). Michael Taylor crashed into the wall trying to make the catch and the ball rolled several feet away.
Shortstop Leury Garcia finally retrieved the ball and threw it in as Gomes slid into third.
Michael Bourn was intentionally walked before Eric Surkamp issued an intentional walk to Lonnie Chisenhall that loaded the bases. Surkamp battled back from a 3-1 count to strike out Jason Kipnis before Murphy, batting for Roberto Perez, delivered.
C.C. Lee (1-1) worked out of a bases-loaded, one-out jam in the 10th.
The Indians, coming off a series in which they lost three of four against Detroit, are six games behind Kansas City in the AL Central.
Chicago ace Chris Sale allowed one run in six innings, lowering his league-best ERA to 2.09.
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Indians rookie T.J. House matched Sale, giving up one run and striking out seven in seven innings.
Carlos Santana's RBI single in the first gave Cleveland the lead, but Alexei Ramirez's third-inning homer tied the game.
Ramirez left the game after being hit on the left foot by a pitch from House in the fifth inning.
Jose Abreu extended his hitting streak to 14 games with a first-inning single. He added a two-out triple in the third.
Chicago has dropped 19 of 27, including six of seven on the road.
Sale allowed five hits, with three coming in the first inning. Tyler Holt led off with a single, but Jose Ramirez bounced into a double play. Michael Brantley singled, stole second and took third when catcher Tyler Flowers' throw went into center field.
Sale struck out five, walked one and threw 98 pitches. Because of a lack of run support, the left-hander has one win in his past seven starts. And that came against Detroit on Saturday when he struck out 13 in seven innings.
Chicago first baseman Paul Konerko, who will retire when the season ends, has a broken bone in his left hand and is sidelined indefinitely.
