Quickest out the blocks, Palace caught United cold after just 11 minutes, as Andros Townsend’s deflected strike nestled into the top corner.
With Mourinho demanding second-half improvement, the visitors fell two behind just three minutes after the break, with Patrick van Aanholt doubling the Eagles’ lead.
Chris Smalling hit back almost immediately, planting his downward header beyond Wayne Hennessey before Romelu Lukaku swivelled to fire past the Palace keeper to restore parity with 15 minutes to go.
United were eventually rewarded for their magnificent second-half showing, with Matic’s powerful strike from 25 yards beating the despairing dive of Hennessey.
“I think Palace deserve congratulations, because a match like this is only possible with two teams,” Mourinho said.
“It was a fantastic match. It was a difficult match for me because my team made so many mistakes.
“We played very good quality football in the second half but we made so many defensive mistakes.
“I don’t like my teams to lose their organisation like we did, but it was an amazing match and a very important win for us.”
A lacklustre first-half display was epitomised by the below-par performances of Alexis Sanchez and Paul Pogba. In truth, Mourinho’s men were average all over the pitch and were out-muscled, outthought and outclassed by a vibrant Palace under the guidance of former England manager Roy Hodgson.
With United reeling from an early Palace onslaught, they fell behind in rather unfortunate circumstances in south London.
Christian Benteke fed Townsend on the edge of the box with the lively winger’s left-footed strike deflecting off Victor Lindelof, wrong-footing a helpless David de Gea.
It was the slice of luck the Eagles had been craving and they came close to doubling their lead just five minutes later.
Townsend was again involved, his dangerous corner bouncing off James McArthur before being hacked clear by a scrambling Lindelof.
United were rattled and Alexander Sorloth was next to try his luck, but his left-footed strike was comfortably dealt with by the untroubled De Gea.
However vulnerable they looked defensively, United’s ability to break at pace would soon have the Palace defensive on guard.
Sanchez’s deaf flick almost fell to the lurking Lukaku, who couldn’t quite latch onto the Chilean’s through ball.
Townsend, in particular, was in the mood to take advantage of United’s disorganisation. He picked up possession deep inside the visitors’ half before rifling his shot narrowly over the bar.
Struggling to generate any sort of rhythm going forward, Pogba’s error-strewn first-half performance summed up the Red Devils’ lack of quality going forward.
Mourinho’s touchline expression exhibited United’s abject display, as his side headed into the break with it all to do in the second period.
The Portuguese tactician introduced Marcus Rashford at half time and the England international almost made the perfect impact, jinking into space before seeing his shot blocked.
Mourinho’s angry touchline demeanour became more pronounced when Palace doubled their advantage.
Quick thinking from Aaron Wan Bissaka after Palace had been awarded a free kick, saw Patrick van Aanholt race clear before slotting expertly past De Gea, but United quickly hit back, Chris Smalling planting a downward header beyond Hennessey to give the visitors a foothold.
As United began to lay siege to the Palace goal, Lukaku scrambled the Red Devils level.
The Belgian striker, now on 99 Premier League goals, reacted quickest to Sanchez’s strike bouncing off the crossbar, before composing himself to slot past Hennessey.
It was one way traffic, as United desperately searched for a late winner.
Pogba’s shot was well parried clear by Hennessey, before more last-ditch defending saw Matic’s goal-bound effort cleared away.
Inconceivably, Mourinho’s blushes were spared by United’s player of the season so far, De Gea.
The Spaniard expertly pawing away Benteke’s goal-bound header to the keep the scores level as the enthralling encounter entered its final 15 minutes.
It was rare moment of reprieve for the Palace defence, who were placed under scrutiny again moments later.
The pace of Rashford saw the 20-year-old stride purposefully into Palace territory before releasing substitute Juan Mata, whose shot was blocked by Hennessey.
United wouldn’t be denied however, Matic’s brilliant strike giving Mourinho’s men the victory in stoppage time, completing a remarkable comeback.
Hennessey 6; Wan-Bissaka 7, Kelly 7, Tomkins 8, van Aanholt 7; Townsend 8, McArthur 7, Milivojevic 7, Schlupp 7 (Riedewald 78 5); Benteke 7, Sorloth 8.
Lee, Cavalieri, Souare, Delaney, Rakip, Jach.
De Gea 6; Valencia 6 (Shaw 67 6), Smalling 5, Lindelof 5, Young 5 (Mata 67 6) ; McTominay 5 (Rashford 45 6), Matic 6, Pogba 5; Sanchez 5, Lukaku 5, Lingard 5.
Bailly, Carrick, Darmian, Pereira.