Kammala wannan jumla: Murray ya sanya hannu a matsayin mai kyauta ta Cleveland Indians a ranar 2 ga Disamba, 1993. A cikin kakar wasa ta 1994, ya buga wasanni 108 kuma yana da matsakaicin batting na.254 tare da RBIs 76, 17 gidaje da kuma 110 hits kafin a yanke kakar wasa saboda yajin aikin. A shekara mai zuwa, ya buga wasanni 113 yayin da yake da.323 batting average for 82 RBIs, 21 gida runs, da kuma 141 hits. Murray ya kai 3000-hit plateau a matsayin dan Indiya a ranar 30 ga Yuni, 1995, a Hubert H. Humphrey Metrodome tare da guda zuwa dama filin daga Minnesota Twins jefa Mike Trombley. A cikin 1995 American League Division Series, ya buga 5-for-13 don.385 batting average yayin da yake da gida daya da 3 RBIs yayin da suka shafe Boston Red Sox a wasanni uku. A cikin
1995 American League Championship Series, ya buga 6-for-24 yayin da yake da gida gudu da kuma 3 RBIs kamar yadda Indians ta doke Seattle Mariners don ci gaba zuwa 1995 World Series. A wannan jerin, ya buga 2-for-19 don.105 batting talakawan. Ofaya daga cikin abubuwan da ya buga shi ne guda ɗaya a ƙasan 11th inning na Wasan 3 daga Alejandro Peña don cin kwallaye Álvaro Espinoza kuma ya ba Indians nasara mai mahimmanci. Yana da wasu RBIs guda biyu tare da harbi na gida a wasan 2 wanda ya zira kwallaye biyu. Koyaya, Indians sun yi rashin nasara ga Atlanta Braves a wasanni shida. Lokacin Murray na 1996 an raba shi tsakanin ƙungiyoyi biyu, Indians da Orioles. Ya buga jimillar wasanni 152, yana buga.260 tare da 79 RBIs, 22 gidaje tare da 147 hits. Ya taka leda a wasanni 88 na Indians, yana buga.262 tare da 45 RBIs, 12 gidaje da 88 hits. Indians sun yi musayar Murray zuwa Baltimore a ranar 21 ga Yuli, 1996, don jefa Kent Mercker.