Roy Hodgson's side started off on the front foot with James Tomkins going close before Andros Townsend opened the scoring in the 11th minute, there was chances few and far between throughout the first period with Jesse Lingard going closest but to no avail.
It only got better for Palace as they doubled their lead at the beginning of the second half through Patrick van Aanholt but Chris Smalling cut the deficit in half soon after, the tide changed from there in favour of the visitors with Romelu Lukaku bringing it level before Matić's thunderbolt secured the comeback.
Looking to get out of their sticky situation
With Manchester City having set the gap to 18 points at the top of the Premier League table the only goal left for United was to solidify their second place position ahead of Saturday's clash with Liverpool, Palace on the other hand were in dire straits at the bottom and their desperation showed as they kicked off on the front foot.
The fourth minute corner in from Townsend was flicked onto Tomkins who had managed to make just enough room around the penalty spot, nobody would have expected what came next as the defender went for the spectacular bicycle kick which whistle just wide of David De Gea's far post.
The Eagles continued to push their North West counterparts in the early minutes and were rewarded with the first blood, it was good build-up play from the home side before Luka Milivojević threaded it through for Christian Benteke.
The striker did well to hold up the ball away from Smalling before teeing it up for Townsend, who made no mistake as he curled it into the far corner.
Despite going behind it didn't seem to kick José Mourinho's side into action as their first real chance of note didn't arrive until six minutes from the end, it was patient play from United which ended in frustration until Lingard found space on the edge of the area but his shot was well wide of the target.
Dealing that double blow
Mourinho decided that he was going to go at Palace with the half-time substitution of Scott McTominay for Marcus Rashford, but all plans were thrown out almost instantaneously with Palace doubling their lead.
They managed to break up the quick play with the foul on Benteke, but Jeffrey Schlupp's quick free-kick caught the entire defence off guard as van Aanholt ran through unopposed to slot it in at the near post.
Start of something special?
United were going to need something or someone to step up to drag them back into the clash and quickly, and just six minutes later they managed to half the deficit through an unlikely source.
The original corner was cleared only as far as Antonio Valencia who was lurking on the edge of the area, he did well to take his time before picking out Smalling at the back post with his chipped ball and the defender finished coolly back across goal.
Finally getting back into it
After getting one back United continued to turn the screw for their equaliser, and won't have come any closer as they did in the 73rd minute.
The corner in from Juan Mata trickled through to Matić at the back post, the Serbian hit it straight and true from the tight angle and looked a certain goal if not for an excellent clearance from Benteke which was backed up by the goal-line technology.
Benteke's efforts proved to be in vain however as United finally broke through moments later, their patient build up culminated in Alexis Sánchez's floated effort clattering back off the crossbar and into the feet of Lukaku.
The Belgian still had a lot to do as he looked to create some space, he finally managed to clear some room to hit a clean effort through the mass of bodies and into the corner.
Their hard work was almost washed away instantaneously as Benteke finally managed to get the better of the defender to get a clear header on goal, but it wasn't for the first time that De Gea came to United's rescue to keep it out with an excellent one-handed save.
Sealing it with a touch of class
United smelt blood as Palace heads began to drop but it did seem like time wouldn't be on United's side, but in the first minute of extra-time United turned it on its head in some spectacular style.
It looked all but over as Paul Pogba's effort was deflected away, but Matić still had a winner on his mind as he settled himself before hitting a great effort from 30 yards into the bottom corner for his first goal in a United shirt.