John McGinn’s third goal of the season put Villa ahead in the 17th minute at Selhurst Park when he curled a shot into the bottom corner.
Christian Benteke equalised against his former club in the 32nd minute.
After Wilfried Zaha had a shot blocked, Mitchell chipped the loose ball into the penalty area, with Benteke heading into the far corner.
But Palace, well clear of the relegation zone and assured of Premier League survival, fell behind again two minutes later when Anwar El Ghazi scored from close range.
The Eagles, however, equalised for a second time in the 76th minute when Zaha scored his 11th goal of the season when he cut inside before a low shot was deflected in off Ahmed Elmohamady.
And six minutes from time, Palace’s comeback was complete when Eberechi Eze miscued from inside the box following a cut back from Zaha only for Mitchell to follow-up from close range as the 21-year-old scored his first goal for the south London club.
Victory saw the Eagles climb to 13th place.
Sunday’s other matches will see deposed champions Liverpool look to close the gap to fourth-placed Chelsea away to already-relegated West Brom, with Tottenham trying to bolster their European hopes against Wolves.
Eighth-placed Everton continue their bid for a top-six finish at home to bottom of the table Sheffield United in Sunday’s late kick-off.