The left-back, on-loan from Manchester United, picked up an injury during the William Hill Scottish Cup win over Auchinleck Talbot earlier this month.
He posted a post-operation picture on Twitter today with the message: “Operation went well – will be back stronger for sure.”
Mitchell suffered the same injury at a similar stage last season while on loan at Tynecastle and was sent back to parent club to recuperate.
After three months out he eventually returned to play in his side’s post-split fixtures with Celtic and Hibs.
Speaking last week, Hearts boss Craig Levein was unsure whether the 22-year-old would be able to pull on a maroon shirt again this season.
He said: “Demi has done the same knee and the same lateral meniscus.
“It’s a shame because he was good at the weekend. He had been suffering from a lack of confidence for a while.
“He had got that back and it’s just unfortunate. I feel for him. It’s a setback for him.
“He can’t remember doing anything that caused it but it must have happened during the game. The good thing is he got back just before the end of last season.”