The game got off to a great start for United when Marcus Rashford won a penalty, which was converted by Bruno Fernandes in the 7th minute.
Rashford scored the second goal of the match in the 62nd minute, giving United a comfortable 2-0 lead. However, a straight red card for Casemiro and a 76th-minute goal from Schlupp put Palace back in the game.
Despite being a man down, United's coach Erik ten Hag made some key substitutions, which allowed them to hold on to the lead and secure the victory.
The game started brightly for United, with Rashford's attacking play leading to a penalty in the 7th minute. Fernandes stepped up to take the spot kick and sent Guaita the wrong way, putting United ahead.
Palace had their best chance of the first half just before the break when Schlupp's powerful shot from distance forced a great save from De Gea. The first half ended with United leading 1-0.
In the second half, Rashford continued to cause problems for Palace. He cut inside and curled a shot just over the bar in the 56th minute, and shortly afterwards, Antony had a chance to increase United's lead, but he couldn't find a teammate and Palace were able to clear the danger.
Rashford was not to be denied, and he made it 2-0 in the 62nd minute with a simple finish after a well-worked short-passing move by United.
However, the game took a turn in the 76th minute when Casemiro was shown a straight red card for grabbing Hughes by the throat.
This gave Palace a lifeline, and Schlupp pulled a goal back soon after to give them hope. The visitors poured forward in search of an equalizer, but ten Hag made some key substitutions, bringing on Maguire, Sabitzer, and Lindelof, which helped United hold on to their lead.
The game ended 2-1 for United.
Manchester United's victory over Palace puts them in third place in the league, eight points behind leaders Arsenal and three behind second-placed Manchester City.
Palace are 12th with 24 points. Ten Hag was happy with his team's performance, praising their energy and team spirit. He acknowledged that the last 20 minutes were challenging, but he was pleased with how the team battled and fought to secure the win.
"It's a big team spirit, they stand (up) for each other and we are happy with that."
"The first 70 minutes was all ours, we dominated the game and we played really well, A big compliment for the team, a lot of energy and they played really well in possession and in the defence transition." - Erik ten Hag, Manchester United coach