Tomlin (1-0) started in place of the injured Aaron Laffey after being recalled from Class AAA Columbus and responded with a fine start against the Bronx Bombers, holding them to one run on three hits, walking none and fanning two.
Carlos Perez collected his 10th save despite allowing a pair of hits in the ninth.
The Indians scored on catcher Francisco Cervelli's error and Matt LaPorta added a sacrifice fly in the fourth to give the Indians the lead for good.
CC Sabathia (13-4) lost for the first time since May 23, allowing four runs (two earned) on nine hits in seven frames against his former team.
Yankees third baseman Alex Rodriguez again failed to hit his 600th homer, going 0-for-4 on this 35th birthday.
Celebrity Breakups and divorces of 2014 [PHOTOS]