Franchitti, a Scotsman who has won four IndyCar series, suffered a fractured ankle, concussion and fractured vertebra during the Houston Grand Prix on October 6.
"Franchitti underwent successful surgery to repair the talus bone in his right ankle," Target Chip Ganassi said on its website.
"After a surgery that lasted just over three hours, he is resting comfortably."
Dr. Terry Trammel, Orthopedic Consultant for the IndyCar Series, added: "Tim Weber repaired Dario's ankle and his surgery went perfectly and just as planned. His post-operation X-rays looked great.
"Everything is positive at this point as Dario continues his recovery."
Several spectators were also injured when the 40-year-old's car went airborne and destroyed a portion of the catch fence.
Alex Tagliani will replace Franchitti for this Saturday's IndyCar season finale MAVTV 500 in Fontana, California.