Johnson's scheduled start Monday had been pushed back to Wednesday because of the pain. The decision to put him on the DL came after he felt sore while playing catch during batting practice Tuesday.
"I finally felt comfortable on the mound," he said. "It's very frustrating."
Acquired from Miami in a 12-player trade last November, Johnson is 2-8 with a 6.20 ERA in 16 starts.
Weber and right-hander Esmil Rogers, who recently lost his rotation spot after nine consecutive winless starts, are possibilities to start Wednesday against Boston.
Blue Jays outfielder Colby Rasmus was out of the lineup for a second straight day Tuesday night with a strained left abdominal muscle.