McAllister (2-0) walked two and struck out four and John Axford pitched the ninth for his fifth save for Cleveland, which had lost 18 of its previous 23 against Detroit.
Axford got out of a jam with a runner on third to seal the win. Alex Avila doubled and went to third on an error by center fielder Michael Bourn with one out. Axford then struck out pinch hitter Don Kelly and got Rajai Davis on a groundout to end the game.
Brian Shaw pitched a scoreless seventh, but Cody Allen gave up an unearned run in the eighth when Indians manager Terry Francona called for a replay review on the play at first base on a bunt single by Detroit's Ian Kinsler and the safe call was upheld. That put runners on first and second with no outs. Torii Hunter grounded into a double play and Miguel Cabrera singled to make it 3-2. Victor Martinez followed with a single, but Allen fanned Austin Jackson to end the inning.
Miguel Cabrera drove in both runs for the Tigers. Detroit starter Anibal Sanchez (0-1) gave up three runs - two earned - and two hits in five innings, walking four and striking out eight.
Murphy made sliding catches in right on line drives by Nick Castellanos in the second and Cabrera in the fourth.
Cleveland got a run in the first as Sanchez minimized the damage after walking the first three batters to load the bases. No Indians batter swung at his first 19 pitches before Carlos Santana grounded into a run-scoring double play on pitch No. 20. Sanchez then got Michael Brantley on a foulout to end the inning.
Detroit tied it at 1 in the bottom of the first on Cabrera's RBI groundout.
NOTES: All players on both teams wore No. 42 in observance of ''Jackie Robinson Day.'' Tuesday was the official celebration around the majors, but the game between Detroit and Cleveland was postponed due to inclement weather. It was 42 degrees when the game began Wednesday night. . Cleveland CF Michael Bourn made his season debut after missing the first 13 games with a left hamstring strain he suffered during spring training. Bourn came off the disabled list Tuesday. He was 0 for 4 with a walk and a run scored. . Due to early season postponements, Detroit LHP Drew Smyly - the Tigers' fifth starter - has yet to start a game. He has made two relief appearances and hasn't allowed a run in six innings. Smyly is scheduled to make his first start Friday. . Thursday afternoon's scheduled starters are RHPs Justin Verlander (1-1, 2.57 ERA) for the Tigers and Danny Salazer (0-1, 6.75) for the Indians.