Salazar (1-0), expected to be a power pitcher, struck out seven and walked one. The walk was the only base runner he allowed in the first five innings.
Blue Jays' starter R.A. Dickey (8-10) was charged with the first three Cleveland runs (two earned) as he allowed six hits and walked three while striking out seven.
Asdrubal Cabrera hit a home run in the first inning and that remained the lone run until Bautista's sixth-inning double.
Dickey hit Michael Brantley with a pitch and walked Ryan Raburn to open the bottom of the sixth. He struck out Carlos Santana but walked Mark Reynolds, loading the bases. Lonnie Chisenhall followed with a bloop single into left-center field that scored one run and another came on a throwing error.
Raburn walked ahead of Santana's triple in the eighth for a 4-1 Cleveland lead.
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Michael Sam cut from the Dallas Cowboys