Janssen, 4-1 with 34 saves and a 2.56 ERA last year, was limited at spring training because of shoulder soreness. The Jays originally had said the strain was in his abdominal area.
Right-hander Sergio Santos will assume the closer's role during Janssen's absence.
Blue Jays general manager Alex Anthopoulos told reporters in a conference call Sunday that Janssen first felt the injury Friday night when he was warming up for an appearance against the Mets in Montreal.
Janssen, who told the National Post that he is barred from throwing off flat ground until the tightness clears up, is eligible to return from the disabled list on April 13.
"We don't expect him to be out too long," Anthopoulos said, according to the Post.
The 32-year-old right-hander allowed one earned run and four hits in three innings during three exhibition appearances.
Catcher Erik Kratz was recalled from Triple-A Buffalo on Sunday, a day ahead of Toronto's opener at Tampa Bay. Kratz hit .400 in spring training and had been sent to minor league camp earlier in the week after losing the backup job to Josh Thole.
The Associated Press contributed to this report.