Smith was traded for minor league left-hander Thomas Pannone and shortstop Samad Taylor.
The 33-year-old and 11-year MLB veteran went 3-0 with a 3.28 ERA in 38 relief appearances with the Blue Jays this season.
Pannone, 23, is 8-1 with a 1.96 ERA and 120 strikeouts in 19 starts with Double-A Akron in 2017.
Taylor, 19, is batting .300 with four homers and 19 RBIs over 28 games with Short-A Mahoning this season.
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