Johnson left Sunday's loss in Detroit in the third quarter to receive medical attention.
Johnson's wrist surgery has an 8-to-12 week timetable. DJ will not be eligible to return to the team until mid-November at the earliest.
Arians added that re-signing running back Chris Johnson, who was cut during the preseason, may be an option. Teammate T.J. Logan is now on the injured reserve with the same injury.
Chris Johnson knows the Cardinals offense well because he spent the last two seasons with the team. Cardinals website senior manager Darren Urban believes that surgery could be on the horizon for Johnson.
Johnson will undergo an MRI on Monday, and the results of that test will determine how much time he will miss. "We've got quality guys on our team and it's an opportunity for somebody to step up".
Johnson finished last season with 1,239 rushing yards and 16 touchdowns.
It was the 24-yard reception that the health of Johnson was starting to become a question.
Stafford completed 29 of 41 passes for 292 yards and four touchdowns. Foster made the Patriots roster past year as an undrafted free agent.
Week 1 in the National Football League was a remarkably lackluster performance for running backs - with just 17 rushing touchdowns over 14 games.