Defender Mitchell bundled the ball home in the 84th minute from Eberechi Eze's effort to make it 3-2, after Wilfried Zaha charged towards the Villa box and blasted home an effort which flew in off Ahmed El Mohamady with 15 minutes remaining.
Villa forward Anwar El-Ghazi had made it 2-1 to the visitors in the first half after skipper John McGinn's side-footed opener was cancelled out by Christian Benteke's header.
The result denied Villa a first league double over Palace since the 1980-81 season, when the Birmingham side won the top-flight title.
Roy Hodgson's Palace sit 13th with 44 points, two places behind Villa who have 49.
Palace host Arsenal in a London derby on Wednesday as fans return, while Villa travel to Europa League-chasing Tottenham.
Additional reporting by Reuters.