Anesthesiologists, who are among the highest-paid physicians, have long fought people in health care who target their specialty to curb costs. Now the doctors are confronting a different kind of foe: machines.

A new system called Sedasys, made by Johnson & Johnson, would automate the sedation of many patients undergoing colon-cancer screenings called colonoscopies. That could take anesthesiologists out of the room, eliminating a big source of income for the doctors. More than $1 billion is spent each year sedating patients...