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. He 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.
Jenkins was optioned to the Buffalo Bisons on March 16, 2014, and recalled on April 26. He was recalled for the fourth time in the 2014 season on June 3, and for the fifth time on July 31. Jenkins pitched 6 shutout innings in relief of Mark Buehrle on August 10 and earned the win, 6–5, after a 19-inning marathon against the Detroit Tigers. The game was the longest in Blue Jays franchise history in terms of both innings and time elapsed (6 hours, 37 minutes). Jenkins was optioned back to Buffalo on August 16, and was recalled to Toronto on August 27, 2014. On September 4, Jenkins was hit by a ball during batting practice while in the outfield shagging fly balls. It broke his right hand, ending his season.
Jenkins was optioned to the Buffalo Bisons on March 25, 2015. He was recalled on May 3, and returned to Buffalo on May 8. On September 11, Jenkins was called up by the Blue Jays.