Sanchez (3-5) had pitched well in each of his previous three starts, holding opponents to four earned runs over 20 innings, but had nothing to show for it.
This time, however, the Tigers scored twice in the top of the first inning on third baseman Lonnie Chisenhall's fielding error and a run-scoring single by Alex Avila (3-for-4, 2 RBI).
The right-hander responded by allowing two runs on three hits. He didn't issue a walk and struck out seven as Detroit clinched its three-game series and remained one game behind the American League Central-leading Chicago White Sox.
Jose Valverde pitched a 1-2-3 ninth to collect his 31st save.
Miguel Cabrera connected for his 37th homer in the victory.
Indians starter Justin Masterson (11-14) was touched for four runs (two earned) on six hits over 4 2/3 innings.
Asdrubal Cabrera, Russ Canzler and Chisenhall drove in runs for the Tribe.