Chavez entered in the eighth inning to protect an 11-4 lead against Cleveland. He failed to record an out while proceeding to surrender a pair of two-run homers before being pulled from the game.
The 28-year-old became the latest in a long line of pitchers who have auditioned for a job in either long or middle relief, but failed to get the job done.
"Putting up a zero -- as plain and simple as that," Blue Jays manager John Farrell said when asked what he needs from the struggling relievers. "However guys do it, whether you have a low three-quarters slot, throw overhand, if you throw one type of pitch for strikes or three types of pitches.
"We're at the point where there is opportunity for anyone who walks in here and if guys can't grab a hold of an opportunity and take a job, we're going to find the next guy."
Chavez made two starts for the Blue Jays this season, but surrendered 10 runs in just 8 2/3 innings. That forced a demotion to the bullpen where he assumed a role in long relief but he struggled there, as well.
Right-hander Chad Beck has been promoted from Las Vegas to fill Chavez's spot on the roster. Beck has made three appearances for the Blue Jays this season and has allowed two runs in 5 1/3 innings.