But the visitors hit back soon after Patrick van Aanholt doubled Palace's lead. And despite The Eagles efforts throughout the second half, The Red Devils equalized with 15 minutes to go, before Nemanja Matic hit an injury-time winner on a sensational rocket from long range for Jose Mourinho's men.
Crystal Palace came out quickest from kickoff. They created early chances with Andros Townsend and Patrick van Aanholt both putting balls into the box; however neither fell quite right for their teammates. James Tomkins tried the spectacular with an overhead kick from a corner before Manchester united had their first real opportunity as Alexis Sanchez received the ball in the box but Wayne Hennessey got himself in the way.
Crystal Palace then broke the deadlock just after the 10 minute mark. Christian Benteke picked out Townsend on the edge of the box and the midfielder let fly. The shot took a deflection off United defender Victor Lindeloff on the way through, but it was heading for the back of the net anyway.
In the 16th minute Townsend whipped in a dangerous corner, and it could have been 2-0, had David De Gea not been the right place.
The visitors then stepped up their search for an equalizer. And although The Eagles continued to have their fair share of attacks, United looked more dangerous and had most of the ball.
Jesse Lingard sent a long range shot wide on 40 minutes, as the game became fairly open and end to end heading towards half-time.
Jose Mourinho made a change before the second half kicked off, as his side looked to get themselves back in the game. Marcus Rashford came on for Scott McTominay and the young Englishman almost made an instant impact as he broke into Palace box, but the home side cleared.
Just three minutes into the second half Crystal Palace doubled their lead. Benteke was taken down around the half-way line, and Jeffrey Schlupp took the free kick quickly sending van Aanholt down the left. The left back broke into the United box and calmly slotted past the keeper.
The Red Devils instantly looked for a response as Paul Pogba tried a long range shot from the edge of the area, before Rashford surged into the Palace box and won a corner.
Rashford floated a ball into the box and Chris Smalling should have scored at the near post, but instead headed wide.
Minutes later Antonio Valencia put a cross into the box, and Smalling made no mistake this time, heading home to get United back in the game.
Palace then broke as they looked to retain their two goal cushion but Aaron Wan-Bissaka's shot was tipped over the bar by De Gea.
Palace looked shaken after conceding and Manchester United started to look very dangerous as they searched for an equalizer.
Despite Manchester United's pressure, Palace started to get a foot in the game again. It was very end to end with Benteke and Pogba both with chances.
After the introduction of Juan Mata and Luke Shaw, the visitors really put the pressure on. They were constantly causing problems as The Eagles found themselves camped in their own penalty box.
Benteke cleared off the line from Matic, when it looked like United had got their equalizer.
United then went again and as a shot bounced back off the post, the Palace defense failed to shut down Romelu Lukaku and he levelled it up with 15 minutes to go.
Palace looked to re-take the lead as they broke down the other end, but De Gea made a decisive save low down to his left to keep his side in the game.
The London side continued to look to regain their lead and the game became very open for the last 10 minutes, as both teams searched for a winner, and just as it looked like the sides would have to share the spoils on Monday night, Matic hit a long range shot in injury time to give the visitors all three points.