He suffered a stress fracture of his right ankle in a pre-season friendly and it failed to heal naturally.
"The initial indications are that the surgery has been successful and Jack will now undergo a prolonged period of rehabilitation," a club statement said.
"It is envisaged Jack's rehabilitation will take a period of four to five months."
Arsenal are 12th in the league after six games and have struggled without the midfielder, who had forced his way into the England first team before his injury.
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