Palace made the breakthrough in the 22nd minute courtesy of a close-range finish from Schlupp, before Mitchell netted in the final exchanges to secure all three points for the Eagles, who have moved into 10th spot in the table.
Burnley, though, remain 19th in the division on four points, with Vincent Kompany's side now suffering nine defeats from their 11 matches in the top flight this term.
Burnley came close to scoring inside the first two minutes, but Zeki Amdouni's header from a Johann Berg Gudmundsson cross was just wide of the target.
There was not too much of note in the 20-minute spell that followed, but Palace managed to make the breakthrough in the 22nd minute through Schlupp, who converted from close range after Jordan Ayew had brilliantly wriggled away from Jordan Beyer and delivered into the box.
Gudmundsson had a chance to level the scores in the 29th minute when he was found by Charlie Taylor inside the box, but the attacker headed just wide of the post.
Luca Koleosho was then denied by a wonderful last-ditch challenge from Palace defender Joachim Andersen late in the first period, and the Eagles entered the half-time interval with a one-goal advantage.
Gudmundsson had another chance to level the scores on the hour when he was found by Koleosho inside the penalty box, but the Iceland international again missed the target.
Burnley did have the ball in the back of the Palace net in the 64th minute through Jay Rodriguez, who had made the most of a tame effort from Koleosho, but the offside flag came to the away side's rescue.
Both managers made changes in the period that followed, which interrupted the flow of what was already a stop-start second period, with both sides struggling to show quality in the final third of the field.
Sam Johnstone made a smart save to deny Josh Brownhill in the 80th minute, before the Palace goalkeeper expertly kept out a header from Rodriguez less than a minute later.
Johnstone also had to keep out a header from Dara O'Shea late on, but Palace made sure of the points in the fourth minute of added time at the end of the 90 when Mitchell found the bottom corner following a clever pass from Eberechi Eze.
Next up for Palace is a home game against Everton next Saturday, while Burnley will be bidding to avoid a sixth straight loss in all competitions when they head to Arsenal on the same afternoon.