Jenkins was selected in the first round (20th overall) of the 2009 MLB draft by the Toronto Blue Jays, and was given a $1,359,000 signing bonus. Baseball America ranked Jenkins as the Blue Jays 3rd best prospect going into 2010. Jenkins was called up to the Blue Jays on August 5, 2012, after posting a record of 5-9 with a 4.96 ERA in Double-A.
Jenkins made his major league debut on August 7, 2012, throwing the last 3 innings in a 4-1 loss to the Tampa Bay Rays. Jenkins surrendered only 2 hits, while striking out 2 and not issuing a walk.
Jenkins was called up from the Double-A New Hampshire Fisher Cats by the Blue Jays on May 11, 2013. He was optioned down to the Triple-A Buffalo Bisons on June 1. After a stint on the disabled list, he was optioned down to the Fisher Cats on August 10. Jenkins was recalled by the Blue Jays on August 24.