By dealing Budaj, the Canadiens opted to keep Dustin Tokarski as Carey Price's backup after the 25-year-old played well in last year's Eastern Conference final.
Tokarski would have had to go on waivers if the Habs didn't keep him in the NHL.
Budaj, who has one year left at US$1.4 million, figures to move ahead of Michael Hutchinson as back-up to Jets starter Ondrej Pavelec.
Tangradi previously cleared waivers and didn't appear to be in the Jets' plans.
The 25-year-old brings size and grit at six-foot-four and 221 pounds.
Forward Patrick Holland also went from the Canadiens to the Jets.