Elias went 10-12 with a 3.85 ERA in 29 starts, but left last Tuesday's game at Los Angeles against the Angels after three innings with soreness in his elbow.
The Mariners had hoped that Elias might be able to work out of the bullpen, but changed their minds after an MRI on Monday revealed a strained flexor bundle.
Elias will not throw for at least four to six weeks.
Also Monday, Mariners manager Lloyd McClendon said right-hander Chris Young won't start against Toronto on Thursday. Young is 0-3 with an 8.36 ERA in his past five starts, and allowed four home runs against Houston in his previous outing.